Teach Better Mathematics - Numbers Operations & Relationships | Leah and Matthew Henshall | Skillshare

Teach Better Mathematics - Numbers Operations & Relationships

Leah and Matthew Henshall, Ms

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49 Lessons (6h 51m)
    • 1. Whole Numbers (1 of 11) division by zero and one

      5:16
    • 2. Whole Numbers (2 of 11) Adding in columns

      7:03
    • 3. Whole Numbers (3 of 11) Subtracting in columns

      8:16
    • 4. Whole Numbers (6 of 11) Number Factors

      7:15
    • 5. Whole Numbers (7 of 11) Number Multiples

      4:12
    • 6. Whole Numbers (5 of 11) long division

      9:03
    • 7. Whole Numbers (4 of 11) Multiplying in columns

      18:08
    • 8. Whole Numbers (8 of 11) Composite Numbers

      5:36
    • 9. Whole Numbers (9 of 11) Prime Numbers and Prime Factors

      9:05
    • 10. Whole Numbers (10 of 11) HCF of numbers

      5:46
    • 11. Whole Numbers (11 of 11) LCM of numbers

      4:30
    • 12. Exponents (1 of 9) Representing numbers in exponential form

      9:43
    • 13. Exponents (2 of 9) Representing numbers in scientific notation

      7:41
    • 14. Exponents (3 of 9) Power of Zero Law

      5:00
    • 15. Exponents (4 of 9) Multiplication Law

      6:23
    • 16. Exponents (5 of 9) Division Law

      6:13
    • 17. Exponents (6 of 9) Power Law

      8:24
    • 18. Exponents (7 of 9) Finding Square Roots

      10:19
    • 19. Exponents (8 of 9) Finding Cube Roots

      7:18
    • 20. Exponents (9 of 9) Exponential laws four operations examples

      11:19
    • 21. Integers (1 of 5) Ordering and Comparing

      7:13
    • 22. Integers (2 of 5) Addition and Subtraction

      11:38
    • 23. Integers (3 of 5) Multiplication

      5:59
    • 24. Integers (4 of 5) Division

      6:35
    • 25. Integers (5 of 5) Multiple Operations

      9:50
    • 26. Common Fractions (1 of 8) Simplifying fractions, including mixed numbers

      12:54
    • 27. Common Fractions (2 of 8) Convert Mixed fractions to improper

      3:59
    • 28. Common Fractions (3 of 8) Adding and Subtracting including mixed numbers

      11:58
    • 29. Common Fractions (4 of 8) Multiplying and Dividing including mixed numbers

      13:28
    • 30. Common Fractions (5 of 8) Finding Square and Cube Roots

      7:10
    • 31. Common Fractions (6 of 8) Finding percentages of whole numbers

      5:25
    • 32. Common Fractions (7 of 8) Increase or decrease a number by a percentage

      9:57
    • 33. Common Fractions (8 of 8) Word problems

      12:24
    • 34. Decimal Fractions (1 of 8) Ordering and Comparing

      13:44
    • 35. Decimal Fractions (2 of 8) Equivalent forms

      6:53
    • 36. Decimal Fractions (3 of 8) Rounding off

      11:21
    • 37. Decimal Fractions (4 of 8) Adding and Subtracting

      9:30
    • 38. Decimal Fractions (5 of 8) Multiplying Decimals

      9:22
    • 39. Decimal Fractions (6 of 8) Dividing Decimals by whole numbers

      8:32
    • 40. Decimal Fractions (7 of 8) Dividing Decimals

      4:55
    • 41. Decimal Fractions (8 of 8) Finding square and cube roots

      8:22
    • 42. Rate & Proportion (1 of 4) Comparing quantities of the same kind (Ratio)

      7:55
    • 43. Rate & Proportion (2 of 4) Comparing quantities of different kinds (Rate)

      10:15
    • 44. Rate & Proportion (3 of 4) Sharing a giving ratio

      7:37
    • 45. Rate & Proportion (4 of 4) Increasing or decreasing a given ratio

      6:52
    • 46. Rate & Proportion (1 of 4) Comparing quantities of the same kind (Ratio)

      6:58
    • 47. Rate & Proportion (2 of 4) Comparing quantities of different kinds (Rate)

      9:36
    • 48. Rate & Proportion (3 of 4) Sharing a giving ratio

      9:40
    • 49. Rate & Proportion (4 of 4) Increasing or decreasing a given ratio

      4:57

About This Class

Welcome to this Mathematics Grade 8 course, on Numbers, Operations & Relationships.

The aim of this course is for it to be taught in a 21st century mindset. The course is actually fun, it uses technology, the internet and general knowledge to better equip a student for real life applications.

This course is intended for absolutely age 12-14 years old whom already has a foundation phase understanding in Mathematics. This course is also suitable for any end of middle-school to start of High School (US) or Grade 7-9 (UK) teachers who want to improve upon classroom teaching methods.

This course is packed with 7+ hours of video content encouraging students to answer questions throughout the videos.

This course is intended to be a resource to help students throughout the year, it can be completed slowly and referred to throughout your year of study. The course has been broken down into the following sub-sections, each containing their own set of videos:

  • Whole Numbers
  • Exponents
  • Integers
  • Common Fractions
  • Decimal Fractions
  • Rate & Proportion
  • Financial Maths

Enjoy!

Transcripts

1. Whole Numbers (1 of 11) division by zero and one: Now we're gonna be doing whole numbers. The 1st 1 we're gonna do is we're gonna discuss the numbers a little bit, and then we're gonna talk about division by zero and division by one. Okay, let's first talk about numbers. What is a whole number? Ah, whole number is any number starting at zero. It is a whole number, so it can be a fraction 1234 all the way up to infinity. It is what I like to call. Ah, hold number. It's a thing that you can hold in your hand so you could hold one apple. You could hold no apples, so it started zero. It goes all the way up and you do not. It does not include fractions, and it does not include negative numbers. Then we have counting numbers. It's exactly the same asshole numbers, except it doesn't have a zero. So counting numbers started one and go to six. Counting numbers are also cordon natural numbers, and this is any number you can count with your hand. So think about counting with your fingers. You got it started. One you can't started zero. Then we've got into Joe's an integer is exactly the same asshole numbers. Except now we go to the negative. An integer can't be a fraction again. It's a entire number is not a fraction of a number and it includes zero goes all the way up to infinity and all the way down, and it always has an increment of one. Then we get positive vintages. Positive integers are exactly the same as natural numbers, but we just call it positive interviews. There we get negative vintages is negative one all the way down. Note that neither of these include a zero and then we get non negative integers and that is from zero. Although after positive so non negative integers are exactly the same as our whole numbers just called something different. Then we have fractions and a fraction is a number that would not fit on these timelines like 1/2 5/7, 5/7 negative for over 3 234 over 33. A fraction can be any number like this. We're not gonna talk about decimal fractions and exponents in the section years. Now let's talk about division by zero. Dividing by Sierra is undefined, right? So we're gonna go through a couple of examples, but you always must know that when you divide by zero, it is undefined. There is no answer to dividing by zero. You can write under find a new book. Let's have a look. An example. What is 23? Divided by zero? We're gonna right on de fined because that's what it is. And we can actually write that in our answer book. Excuse the handwriting. What is one of a zero under find? You can write that in your book. What is your of zero? So zero divided by zero is something record into terminals, but you can also write under Final Be just as right that indeterminate means. We cannot actually figure out what this number is. There's no way for us to tell, but it is. The same is undefined, so Division is putting into equal groups for sharing. Let's have a look, at example, to their six piece of cake, and two friends want to share them equally, so we know that six divided by two or out of the six piece of cake. We're sharing it between two friends, and we know that that's going to give us three piece of cake per friend. But what if we wanted to take the six piece of cake and shave them between North frames? Well, that doesn't make sense at all. Six Divided my notes. We're gonna get, of course, undefined. And you can write undefined or does not exist or impossible will give you the answer. But if you remember undefined, we can't figure it out. Let's look at division by one again. I'm a start off by telling you the rule deficient by one is always unchanged. Okay, so what I mean by that 23 divided by one is equal to 23. It is unchanged. What is one divided by one that is equal to one? It's a whole What is 452? Divided by one That's 452. It is unchanged. What is 23.5? Divided by one that is 23.5 and that is the same as saying 23 and 1/2. I've divided by one, but still equal 23 and 1/2. Okay, so division by one is splitting it toe one equal group. So let's look at example there six piece of cake and one person was to share it with himself. So six divided by one is six. It's unchanged. He gets all the piece of cake. Every whole number could be written as a fraction of one. For example, Forever one equals four, which equals 4/1. It is exactly the same, right? So now we've covered whole numbers division by zero makes you're going to go to add and subtract columns. 2. Whole Numbers (2 of 11) Adding in columns: OK, guys, now we're gonna be doing, adding and subtracting in columns, right? This isn't the whole number section. We still got exponents into just common fractions, decimal fractions and lastly rate proportion and financial mess to go. But let's not look at adding and subtracting in columns said We want to add two numbers 223 on and 33. These the ones. So we know there's three ones. These are the tens and these 100. You should know this by now. That's 100 that's 200 plus 22 tens, plus three ones where there's three tens over here and three ones. You should understand this now that those are two hundreds years to tens and three ones. This is how we split up 100. This is the same as 200 plus 20 plus three, right? So we want to add these in columns. Economist. These roads going down like this. So if we want to say 223 plus 33 right, so we set it up exactly like those not importing important with adding in columns. We always gotta push the numbers to the side here, right? It's always gonna be along the right If there was a decimal year, this, we can assume, is the decibel. So if you have a number, say this was zero is the same as a 223.0 and 33.0. We always got to line up the decimals. That's extremely important. So let's add the ones that's three plus three. We know that is equal to six ones, right, two plus three. We know that is equal to five tens and in 200 plus zero, we know that he is equal to two hundreds. So we get a chance off 256. And that works well now. It's also nice to known, adding columns that this is exactly the same as this. If we switch it around, it still works as long as we keep the columns where the decimals are lined up perfectly so that three plus three is six. Three plus two is five and two, or zero plus two is two. On that is 256. That works perfectly now. What if eso these the winds? There's the tens and that's the hundreds. What if we would have changed this to 223 past 38. So it's changed this now, right? These are still out ones, obtains and are hundreds over here. So let's make those 223 has 38. We still keep it in. Our columns remain. We have a decimal place here, 38.0 223.0. So we line up the decimal players, and what do we do now? We go three plus eight. So three ones plus eight ones. What is that? 3% is equal to 11 right now. 11 we contradict causes two digits, but 11 is the same as 1 10 +11 right? Okay, so let's put the 11 year and it's add another one over there, and that's our one. Very good one. So that's one plus two. That's three. Plus another three or six, right and in two plus zero is, too. And we get 261 and that is our answer. Okay, so we added that 10 over there. Let's try a few examples. What's two plus nine? So 242 plus 39 to pass nine we know is 11. So it put the one day and we carry one. So one plus fours. Five plus another three is eight plus two is two. I cannot let's look, at example to have 272 plus 37. So we know we can say two plus seven is equal to nine. Seven past three is equal to 10 right? Oh, this is remember, 70 plus three. The 70 plus 30. Sorry. Seven triples. 30 is 100. So that is the same as even taint eins or 1 100 So we're gonna go and put the zero here and put a one above there, and we're gonna add that 100 year, so two plus one is equal to three, and that is equal to 309. Great. Let's have a look at this. Next example. 273 plus 27. Three past seven. Is he got a 10. So we're just gonna write zero we're gonna carry the one to above their one per seven is is eight plus another two is 10 again. Okay, so we're gonna just leave the zero. They're gonna carry the one over one past two is equal three. And there's Answer 300. Let's have another look at this example. Now we have some decimals that are different cause we've kept our decimals in line, so it's still fine. We could still do this. Remember, there is a zero here, so literal draw that in two plus zero is too. So now five plus seven is 12. So we put it to their we put out decimal. Don't forget your decimal. We add a one over here. One plus 23 plus another 36 We're not carrying anything. So one plus one is two. 26.22 Well done, guys. Getting the hang of this. Let's have another look. At example. Four plus seven is equal to Elevens. We're gonna go one name and over one of you here eights. Press one is nine. Has another three is 12. So we're gonna put it to year. That doesn't matter. That we I had the smaller number. The top one plus fours. Five plus another three class A 1 1000 13,000 521 Well done, guys. Great. Let's have a look at this. big example. Nine past seven is 16. So put a six. We carry a one one has 56 pass. Another four is 10. So we're gonna put a zero here, carry the one This we're gonna have 12. So we put a two year carry. The one now we have four plus minus 5678 Old isn't eight. So we're not carrying a one. Three plus three is six. Great. One plus three is full. Great to pass. One is three. Two plus zero is too. And one plus zero is one. Make sure you keep your column straight in a big example, like does it can often get a bit confusing. 3. Whole Numbers (3 of 11) Subtracting in columns: OK, guys, Now we're gonna be doing adding and subtracting in columns. Now it's everyone to subtract two numbers 453 and 41. So we want to take 41 away from 453. We've got out. Once we got obtains, we got hundreds. So we got 453 minus 41. Right? So the only difference years, we gotta miners. So let's do this. Three minus one is to five. Minus four is one full minus zero is full. So we have 412 right? What you can do is try and add this back together. Check that 412 Fast 41. Remember to keep the commerce and the decimals in in line and see if you can add that together. So we're gonna get a three over here. Two guys, one a one person for is five and a four plus zero is four. And that's 453. That's exactly the same. So it works. We gotta drive. Something very important to note is that 453 minus 41 is not the same as 41 minus 53. You conscious. Switch those around. It's very important to remember this. Now. What if we add this to fully four? We make this of four. What if we say 453 minus 44. So we're 453 minus 44 Right in adding we added units so we would add a unit to the 10 but in subtracting, we've got to borrow units. We can't say three minus 43 minus four equals negative one. It doesn't work. So we have to borrow a unit from the 10. So what I'm gonna do is I'm gonna cross that five outs and make it a full. So we mine ist 10 from the tens column and they were gonna add the 10 yo, And what I do is a drawer, little one, and that is now 13 over here. So 13 minus fall is equal to nine. Four minus four is equal to zero and four minus zero is equal full. So we have 409. Try and add those together to see if it works. Okay, let's do some examples. What's 249 minus 39 nine minus nine is zero So we have a zero full ministry is one two minor zeros to turn around 10. You can always check thes by adding these two numbers together together to see if you get that number. Okay, what's 270 to minus 37 Right. We've gotta borrow number here, because what's to minus seven? We can't do that. That doesn't make sense. It's not. It has to be a bigger number at the top. Right? So we're gonna borrow one. We're gonna cross off that seven. Make it a six. We're going to say 12. We added the one and 2 12 Minus seven is five. Great. Six minus three is three. That's great. That's bigger to Monday's ears, too. Great. Check that aunts again. By adding it together. It's good practice. What is 233? Minus 227 Right. If you just looked at that, you could probably toll that if we were in 227 plus six would get 233. So the answer should be six. Let's check that. What is three minus seven? We can't do that. So we need to go and borrow one. So we take that and that becomes a to And then we add our one new 13 minus 76. Tu minus two is zero two months to zero. Again. Saw answer is six. Well done, guys. Let's try this example 1240 to minus 157. What is to minus seven. We call into that. We need to borrows. We got three. We add a one 12 minus seven is five remained. We only can never borrow from the one next door to it because it's exactly 10 times bigger . So what's three months? Three is euro. What's two months? One is one and one. Well done. What is 1240 to minus 453. Okay, Tu minus three. We can't do. We need to borrow one. We make that a three. Make that a 12. What's 12 minus 3 12 Ministry is nine now. What's three minus five? Oh, no, we can't do that again. We'll have to borrow and add to the three. So let's borrow from you and make that a one. Let's add a 12 or three, making it 13 13 minus five is eight. Well done, guys. Now we have one minus full again. We can't do that so we can borrow this one year, making us in Lourdes adding, Are one over there. And now we have 11. Minus four is seven. And then because it zero here, zero minus zero is zero 789. Well done, guys. Practice examples like this. Let's have a look at this big example. A big number minus a big number. Right. So what is four minus seven? But this just show you could make his numbers Bigas you wanted and it could still work. What's four minus seven we can't do that is to small. Let's borrow four. So I'm gonna borrow that for we making that a three on. We're adding a one there, So 14 minus seven is seven. Right? What's three minus four? I know we can't do that. So is borrow one year. So we make that 2 to 1 and we add this 13. What's 13 minus 49 Okay, one minus nine. We can't do that again. We have to borrow again. Silvestri, we add the one at 11. 11 minus nine is to three monastery are great. We could do that. That equals zero three Ministry again is equal to zero. Remember to keep very careful with your columns. What's one minus three? We can't do that. We're gonna have to borrow one at a one 11. Minus three is AIDS. One minus one is one Tu minus heroes two and one, monasteries one. And that's exactly how we do subtracting in columns. Okay, you're noticed. In all of these examples, the biggest number was on the top, and the negative was the smaller number. Most of your questions will be like this, and you don't need a stress too much when it's not that it will deal with it in future years. But right now, all you need to know is that if the bigger number was negative at the bottom, it would just give you a negative answer. Let's just have a look. A quick example. What if you wanted to go seven minus nine? Now this might seem difficult because seven is smaller than known. Right, But let's say seven minus known right. We've got a zero on our number line. We have our seven on our number line, and we're gonna minus 97 minus one minus two, minus three minus four, minus five, minus six, minus seven, minus eight minus nine to get minus two. So what we can do here is we go seven, We're going to switch it around. We're gonna take this nine to here. And when we switch around a negative number, we've got to drop the negative sign. So what is nine minus seven is too. But the answer is negative. Two. And that's what we got every year, right? Let's have a look at 53 minus 64. Let's switch it around. And what do we do if we have to switch it around? We have to drop the negative sign and we're going to go 64 minus 53 for minus three is one great and six minus five is one. And that's onsen. Negative. 11. Remember, You must not forget this negative sign is absolutely crucial. And that's to do a subtraction and adding that we did in the previous video Great 4. Whole Numbers (6 of 11) Number Factors: okay, We're gonna be talking about number fact. We're still in whole numbers and we're going to deal with what is a number factor. This is a numbers, operations and relationships. So what is a number factor? Factors are the numbers. You multiply together to get another number, right? Factors are the numbers. You multiply together to get another number. Let's have a look. What are the factors of AIDS? What are the numbers? You've gotta multiply together to get to eight. Okay, let's have a look. Eight times one is eight, four times to his aides. So 814 and two are your factors off eight. But now is six. Effect of eight or is 16 effect of eight? Because if we win six times eight divided by six, we would get eight. If you put that in your calculator, you see Nolan's his aides. So could we say that those are factor of aids, or what? About 16 times 1/2 that equals eight as well. This can't work for it To be a factor in must be an integer. This is extremely, extremely important. So what about negative items then? So these are not factors? Is not forget. So what about negative numbers thing? Negative. Eight times they could have one is equal to eight. Negative. Four times negative two is equal to eight. South factors of age are negative. Eight over there. Negative. Full negative too. Negative one. And in 1234567 years. Now, most of the time when we talk about factors, we only going to concern ourselves with the positive numbers. But just so you know, negative numbers can also be factors as long as it's an inter Joe on. We remember that interject is a positive or negative number that is not a fraction or whole number. Okay, so what are the number? Factors off seven. So seven times one is equal to seven. We know that. Let us try and find another one. What about 07 times What? What about three times to know that equals six? That's not affect. Er, what about four? No, I mean, if we were 3.5 times to that equals 7 3.5 times too. But that's 1/2 as London Inter. Joe. So the only factors of seven or seven and one. Okay, Okay. Let's clear that. What are the factors of 12? Okay, what are the factors of 12? Let's try this 12 times one. Now what I want to show you here is if it is even, you can always harv it to find a factor. So let's bear in mind that that cannot be effective. 12. That is more than half than half of 12 right? So what? I mean by that effect of 12. If we have 12 it's half 12. 12. Divided by two is equal six. So that's easy. So we can say six times two is equal to 12. Brilliant. They can never be a factor off a number that is between half and the number So 78 1911 can never be effective. 12. So we can always harv it if it's even. Okay, so let's go down is five effect of 12 5 times three is 15. So five times three is 15 is too big and five times two is 10 and that's too small. So 12 is in the middle here. That doesn't work. What about four? Oh, yes. We know. Four times three. Is it going to 12? Okay. And then we got 23 and four. So we if we wanted to keep going down, we could say three times four is equal to 12. And we could say that two times for as six is equal to 12. But we don't need to do this because we've got them already. Some factors of 12 of one to three full six and 12 right? Let's care that What are factors of 24? Again? Let's divided by two. So we first noted 24 times. One. That is a factor for every single number cause that equals 24. Then let's Harvard again. 24 divided by two is 12. 12 times two is equal to 24. What about three. Let's try three year. Is there anything Time Street? I can get us 24. Three times six is 18 3 times savings to anyone. Three times eight is 24. Okay, brilliant. Let's try for Is there anything times four that will get us 21. I mean, 24. Let's think about this. Six times four is 24. Brilliant. And what about five? The only numbers left off five times five, and that equals 25. So the factors of 24 1 23 full. Six eight 12 24. Great. So now what are the factors of 100? This sounds like it's getting a little bit more out of hand, so we know that the hundreds times one is 100 brilliance every time. That will be an answer. We know that we get It's even, so we can Harv it. So let's do that. So it's 50 times two is equal to 100. Great. Three. Know that word work? Because it's three times 33.33 That word work. Let's try four. Oh, that will work 25 times four is equal to 100. What about five? What times five equals 120 times five. What about six. So the way you can check the Knicks numbers ago. 100 divided by six. See what that equals. 100 divided by seven. What is that equal? 100 divided by eight is also not gonna work for us. And you can put these in and check them for example when 100 divided by six about 16.67 and that is not a whole number, is not interject, so we can't use it. Seven will give us a non. And Incheon, So eight. What about nine? 10 are tinkered work. What about 10 times 10? Brilliant. That is equal to 100. And now that we've made the same numbers were the lowest and the highest Admit there can't be any more numbers. So we know the factors of 100 or 1245 10 2025 50 and 100. Well done, guys. So that's what we're gonna do for number factors. 5. Whole Numbers (7 of 11) Number Multiples: Okay, now we are whole numbers, and we're going to do multiples of whole numbers, right? So what is the multiple of a number? The multiples are the numbers we get after multiplying the number by integer. Right. So if I said what of the multiples of three? So remember when we were doing factors? There was always a set amount of factors. For a number, you couldn't just keep finding more. So for multiples, their infinite amount of multiples. We confined for a number. So what are the multiples of three? Three times one is three great. Three times two is six grades. Three times three is nine. You see, we multiplying it by an Inter Joe, we keep in three consistent or we keeping three constant. So that's nine three times zero. Because zero is an integer Oh, is equal to zero. That was wrong. Be careful. Way is your, oh, three times negative One is equal to negative three, three times negative. Two is equal to negative. Six, three times negative. Three is equal to negative. Nine remaining negative. One negative to negative. Three are in Tages, so I don't need it tonight. And this goes on all the way. Apus fires You want to go and that goes negative all the way as far as you want to go, right? So remember, it has to be an integer. Don't forget that. Let's look at this. Next question is seven on multiple off. Three right. Easy way to check that is, Go seven divided by three and we find that in equals two and 1/3. So it is not a multiple of three because it is not an integer. Over here is 942. A multiple of three. That's easy. We just got 942 divided by three. You can put this in your calculator, and that equals 314 there. No remainders. So, yes, it is a multiple off three. So 942 is a multiple of three. Is how we say that answer right is 12 on multiple of full. So that's easily said. 12. Divided by four. And that equals three. Yes, so 12 is a multiple of four because it's free, say, four times three. It will give us 12 rights. Another multiple of four is four times one four times four times two four times three we have there and all the way up and down is five, a multiple of 15. Again, Let's go 15 divided by five and that is equal to three. And great. It is a multiple off three is five a multiple of this massive, enormous number, right? So if it ends in a zero or a five, five and 10 are always multiples from so this is signed very useful to write down. And remember, if a number ends in a zero or a five, five and 10 the numbers five and 10 are always multiples of the number. Okay, is to a multiple of this massive numbers. If it ends in an even number two is always a multiple. There's another always with learning is 17 a multiple of 578. The only way to figure out this is go 578 divided by 17 and we find that it gives us a whole number. If we said 579 divided by 17 we would get 34 remained a one, and that is not a whole or air multiple off 17 so we know that if we were in 17 minute times. 34 that would give us 578. So it is a multiple? Yes. So these are multiples. These answers are yes. Great. Let's move on. That's multiples down. We're gonna now move into composite numbers. 6. Whole Numbers (5 of 11) long division: Okay, Now we're gonna be doing some long division. We still only working with whole numbers. And we're gonna show you this method you can use for long division that you've probably learned to. We're gonna go over some of the rules for dividing in type of size number you want. So let's just divide what is four divided by two. We know that that is too silly. One. What is five divided by two. Well, we know that that is, too. And then there's remainder one. That's our remainder. 10 that's 2.5. Okay, so what is 21 divided by seven, I guess. And are we going to donate this in long division format? So we put this little brackets half bracket and her line, and we say it's 21 divided by seven. So to read this out loud would say what is 21 divided by seven. And we put our answer, which we know is three at the top as three. Are we gonna go through out to do this though? So this is what we're gonna use to denote long division. We got to say what is? 224 divided by seven. Okay, so when you would do multiplication addition is attraction. You started at the right, and you worked Lift. Now we're gonna start at the left and work right for long division. So what is seven into two? Seven into two goes zero times because seven cannot be divided into two. So we gotta put a big zero. Their what is zero time 70 times seven is zero. Right. So you draw a line across year across two of the numbers and we say to minus zero or 20 to minus zero is 22. I can, and they were going to say what is seven into 22? Well, we know seven goes into 22 3 times because we know seven times three is to anyone. So it different doesn't go in four times because seven times 22 7 times four is 28 to 7 times three is equal to 21 and seven times. Fall is equal to 28. You need to know your times table to do long division cervical ago. Seven into 22 goes three times. Three times seven is 21. Okay, Now we draw this underneath the four as well. We say what is 20 to minus 21 0 to minus one is one two months to zero. We bring down our four and we get a 14 over here. Okay? Suggestive. You asked how to seven going to 22. Now we're gonna ask how many times? A seven going to 14. And we know that goes and twice because we know that seven times two is equal to 14 and then 30 tours are answer without any remainders. Great. And so we had a line there. We have a line there that we draw. OK, now let's look at this example. How does how many times? A seven going to 226 rise? So seven goes into 20 times zero times seven is zero and we bring the 22 down to get 22. Seven goes or we bring the to minus zero. We got to and we brought this to down seven. Goes into 22 3 times seven times three is 21. We're going to draw a line across here. We say 20 to minus 21 is equal to 01 We bring this six down and we get a six. So how many times the saving rate of 16? That is twice. And we're gonna have a remainder off to cause we're gonna go to times seven is 14. We can't bring anything down besides the remainder. So 16 minus 14 is too. So we know that that becomes a remainder of two. So 32 remainder two. Let's have a look at this example. How many times? 21 going to 2222 6 How many times has 21 going to two? We know that. That goes in zero times 21 times zero is zero. We bring up two to minus sorters to We've been out to dinner. We get 22. How many times? The 21 going to 22. We know that it goes inwards. One times 21. He's 21. So we minus these two numbers 20 to minus 21 we know is gonna equal was one. We bring us six down. What is six we got yet? How many times is 21 going to 16? It can't go into 16 so it must be zero zero times 21 is zero. We're gonna put two zeros in here and then 16 minus zero is equal 16. So we know that that is our remainder. So it's 10 remainder 16. So let's do that. What if we went 21 times 10. 21 times 10 you know, is equal to 210 right? And then what if we went? Plus 16? We know that 210 plus 16 is equal to 226 and there's answer it can't just cared the writing , for example. Three. Now, this example seems a bit out of our depth, But we could do it even though it looks so big. How many times? A stream of 40 going to 30 times zero times through under 47 is zero three minus zero is three. We bring down our five and we get a five. How many times? A stream of 47 going to 35 0 zero times 247 we know equals zero. So you put on a zero here we pull it across. How many times the street 135 minus zero equals 35? We bring down our zero here and we get a zero. How many times does 347 go into? 350 are It? Goes in once one times 247 347. That's quite easy. And I always say 350 minus 247. We know is gonna be borrow one. We take the full. We added one over there. 10 months. Seven is three four miles. 403 months. 30 We got a bit are three down. Don't get confused. The columns we had our three. How many times? A stream of 47 going to 33 0 zero time Syrian or 47 is equal to zero. So we're gonna minus these 333. We're gonna bring down our to Here it comes. How many times has 347 going to 332 0 times. So add a zero. That time zero we know is gonna give us your own. We do this again. 3 330 to minus euros. 332. How many times we bring the six down. So how many times has 347 going to 3326 right? Let's think about this. 347 times 10 is equal to 3470. Okay, so we can see that is not going in 10 times. But it most likely is gonna go in nine times because that is less than 347 less. So we had a minus 247 from those. So what is 347 times nine? It's just simply that minus 347. And we can work that are quite easy. That's we borrow one, we go. Six says 10. That's 36 minus four is two for minus three is one three minus zero Is three a coup. This number is smaller, so it must go in nine times. So now we've just worked out what nine times 347 is and that is 3123. So we can write that day because this is now continuing there. 3123. You must use your cochlear if you need. Now we simply got to minus thes two to figure out what I remained is six minus three is three two months. 203 minus 21 is 23 months. Three is zero. So the answer is 1000 and nine. Remainder, 203. So that's how easy it is to long division. Big numbers. Keep practicing it. Make up your own ones and they multiply them back together to see if your answer was right . Well, then. 7. Whole Numbers (4 of 11) Multiplying in columns: Okay, Run. Now we're gonna be doing multiplying in. Columns were still in the whole number section. That's a great trick to learn. And you can multiply big, big numbers together as big as you want. So we want to find the multiples off to number. We want to multiply 223 and three. Those are ones. Obtains are hundreds is not forget. There's 200 years to tens day there's three ones. And here there's just three ones. So when we have 223 times three or multiplied by three remember, we have to keep our units in the same line together. It is so important that we do this. If we start moving them around, it won't work. Remember, a good rule is to line up your decimals, line up your ones, line up your tens, line up your hundreds. Now, in this example, we go three times three is equal to nine. Three times two is equal to six and three times to again is equal to six now in multiplying . Unlike subtracting, you can keep these numbers. You switch them around into two equal the same thing. So 223 times three is the same as three times turning 23. So now this example, What we did here we win this number times this number, this number times this number this time number times this number in this example We do it completely different explains, or in a few minutes. But it's ancient. We're going. This number is 00 So we're going three month times three or multiplied by three is 93 times zero is zero. You know that time 003 times here again is zero. Don't forget that. Three times 00 two times three. What we do then, is we add our magic zero, which I will explain to, you know, there we go. Two times three is six. Two times zero is zero. Two times zero is zero. Okay, Now we on this first row, second row, third row. We're gonna add to magic zeros. Remember to keep your columns straight. So we're gonna go to times three is six two time 00 two times 00 Now we're gonna add all of those together. Right? Nine plus 0.0 is nine zero plus six plus 06 00 plus 66 009 and zero has nothing is 0 669 the exact same Answer this work through those so multiplying in columns this practice, a couple of examples, what is three times to six. So you start there today. What is three times for that is equal to 12. But the contract 12 year. So we're gonna carry the 10. So we're gonna go three times to Is he going to? And we're gonna add this one over here when we still gotta multiply now, three times to his sex plus one, we not multiplying it just plus one. So three times to six plus one is seven. And that answer is 726. Let's try that again. What is seven times three? Seven times three is 21. So we only add in the one, and we put it to over here. What is seven times seven? Seven times seven is 49. You've got to know your times tables. What is 49 plus two 49 pastors, 51. So 51 we add a one and we add our five. OK, so now, because we're on the last one. What is seven times to that's 14 class 5 14 plus five is 19 so we can now just write 19 year, and that is our answer. 1911. Let's have a look at this massive example. What is nine times five? Nine times five is 45 3 admiral five and we add, Afford the top. What is nine times 49 times forced 36 36 plus four is 40 So add a zero and we take our tens , which is our 40 across. What is nine times three is 27. We take 27 plus full was 27 plus 4 31 So 31 and we put out three of the top here nine times to 18 plus three is 21 were at a one year, enough to to the top. What is nine times one is nine plus two is 11. Be so careful with your columns. This is where you can get very mixed up in the big numbers. It's useful to remember that in your times, tear in your nine times table one times nine is equal to nine. Two times nine is equal to 18. Three times nine is equal to 27. Four times nine is equal to 36 on. If you don't know, this is a very useful thing. All the digit in your nice times table up to nine where nine times nine is anyone. What always add up to nine. So what's eight plus one? A. Plus one is nine. His nine. What is three per six? Three past six is equal to nine. What is two plus seven to plus seven is equal to nine. One past eight is equal tonight, and it's a great way to remember your nine times table. Okay, that was just some extra information. Let's have a look at the example off. 213,345 times by zero What is your A times five. Don't drive 50 times 50 What is your times? 40? What is your times? 300 times 200 So we get zero when we times about 10. Okay, I'm just telling you this for a certain reason that you're going to see now. Let's not try multiplying two numbers 223 times 13 rides. That 223 times 13 is exactly the same as turning 23 times three plus 223 times 10. That makes sense. That is exactly the same. Just have a look carefully. It does make sure you understand. So if we got 223 times three, what is that we've done? This is easy. Three times three is known three times two is six, three times to six right now. What if to this one, we plus this number right? So it always in and a zero every times anything by 10 or 100. If it has a zero, then it always end in a zero. So 223 times 10 is equal to 2230. That's easy. Just at a zero on now it's very easy, so we can just add the two together, we can say 669 remain. We got it ended at the because 669 plus 2230. I haven't drawn this very neatly, but you can see there's a zero day so ninth of Syria's nine six mastery is 96 Press to don't get confused over the columns is eight and two. Plus two is to our answers. 2899. It's tried like this. We build up our multiplication three times. Three is 93 times 263 times to again is six. Now. If I went one times three, my answer would be three. But I'm not going to do that because they has to be a magic zero. Yo, So what is one times three is three. What is one times two is two. What is one times two? Again, that's too. And you can see that is exactly the same as the Scipio. 9096 pastries nine A plus two is eight and two plus twos to or to play zeros, too. And that is the exact same own. So we must get on Magic zero. Let's go through some examples. 242 multiplied by 33 3 times three is equal. Six, three times to again is 12. So we're gonna add up to an ana one over here. Rights three times two is six plus one is seven can. Now we're gonna add our magic zero three times to six. Three times four is 12. Okay, so we're gonna add up to here. Keep our one again. I'm making new row ones over here. Three times two is six. That's another one is seven. Okay, so now we're gonna add these together. Draw a line at a plus sign. Six plus zero is six. Two plus six is eight. Seven plus two is known and seven past zero or zero cost 77 and that is on. Said is trying that again? 273 times 17. What is seven times three is 21. We had R one and r two. Seven times seven is 49. 49 past twos. 51. We had a one. We had our five seven times two is 14 plus five is 19. Weaken straight away. At our 19 we gotta add a magic zero one. Time Street is three one time. Seven a seven and one times two is two. We got a plus. He's altogether one times one plus series. 11 plus tree is for nine. Past saving is 16. So we gotta add a six and a one one plus one is to pass. Another two is full. So 4641. Let's do one more big example. What if we go? 12345 times 79. Okay, what is nine times 5 45 were at our five and I fall. What is nine times four is 36 36 plus four. You might recognize this from previous the previous slide. 36 plus fours. 14 were now zero. We had our full nine times. Three is 27 plus Force 31. We had a one way at all. Three, nine times two is 18 plus three is 21. We had up to nine times won. His nine past two is equal to 11 and we add our 11. Okay, now we go and we add our magic zero. We say saving past 57 times five is 35. Way out of five. Now we're gonna make another row and we got past three year seven times four is 28 plus three is 31 where the one we are three. Seven. Remember to keep your rose. Don't get confused. Seven times three is 21 past three is 24. We put up to every year for 27 times. Two is 14 plus two is 16. We had our six and No one is not. Forget arose seven times. One is seven plus one is eight and we add out eight. Right now. Don't get confused of the columns. Here we have five plus five plus two plus five plus seven and past nine, and that is our answer. Okay, let's try this example. When we have three at the bottom right, you're gonna have to add in to magic zeros because there's three at the bottom. So there's three rows here. What is three times to six? What is three times for 12? So we can only add the two and add the one. What is three times two is six plus one is seven. Adam Magic zero. What is three times two is six. What is three times four is 12 rights. We had a one and we add up to for 12. What is three times to six again. Plus one is seven, you know, said those are exactly the same numbers with the magic Zero because these are exactly the same at the bottom. The threes. So now we're gonna add to magic Zero because we're on the hundreds now and we know hundreds of two zeros behind them. What is four times to its eight? What is four times four is 16. We add the six we add our one on our third road. What is four times to eight pass one is nine. And this now we add together. Six plus 0.6 to plus six is eight seven plus twos. Nine plus eight at 17 at a one eight plus six is 14. We had a one and 9% is 10. And that's onset draw era underneath. Let's have a look at this big example Year now we have four here, which means we're gonna have four columns. Here. We here. We only had three. Now we're gonna have 1/4 1 over there. So let's say nine times five is 45. So we had five and we had a full nine times. Force 36 has full. This is again 40 nine times three is 27 past four is 31. We had a one and are three nothing. Times two is 18 plus three is 21. We had R one and R two nine times one is nine plus two is 11. We add our 11 no way at all Magic zero four times five is 20. We add our zero on we added to here, remember, you can draw a line if you want to make it easier. Four times four is 16 plus two is 18 to add our eight and we had a one from the 18 four times three. Remember your columns if you need four times three is 12 plus one is 13 at a three and no. One four times two is eight plus one is known 44 times. One is four plus. Nothing is just a fall. So now we've added to magic zeros here and here. Get what is five times 5 25 Sorry, this five or he has been Macy And then we add it to the top and we've got out to magic zeros. What is five times for its 20 plus 2 22 We add up to at the top What is five times 3 15 16 17 We had all 17 and I won. What is five times to its 10 plus one is 11. We had a one and I want a top. What is five times one? It's five past one is six. Now. We're gonna in the thousands for the seven. We know that at seven is representing 7000. And there's three zeros there. So we've gotta add 123 zeros. What a seven times 5 35 Now we've going up now into the actual columns here. What is saving times five is 35 to add a five plus R seven seven. Okay, we can put it over there. I mean, are three we can put all three over here. What is seven times four is 28 plus 3 28 29 30 31 This ad history again. What a seven times three is 2122 23 24. They had a four we add up to for 20. What is seven times two is 14 15 16. We had us six over there and I won on what, seven times? One is seven plus one is eight. We add our eight year now we're gonna add this massive thing together. Five plus 0.5 zero plus zero plus zero plus 00 1% is known plus a five. Remember to five is not 14. We had a full and No. One 11 is two plus three is five. That's 27 plus five is 12 so add up to and I won. One plus one is two past nine is 11 plus seven is 18 plus one is 19. We had a 11 plus one is two plus. Four is six plus one is seven plus. Another four is 11. We had a one here we have one plus 67 plus another six is 13 plus 1 to 8 is nine, and that is our answer. And that's our answer. Check it on a calculator if you want. Right. So there was a bit of a big, messy question, but it's stuff you can practice and can do. So give yourself a taste making as big as you want. It was a great way to practice your times table. Great made a practice these and they check it on a calculator afterwards. So that was our multiplication. Well done, guys. 8. Whole Numbers (8 of 11) Composite Numbers: again. Now we're gonna do composite numbers and talk all about what? Conversant numbers. And we're gonna figure out that composite and prime numbers are the opposite from each other. Let's go into it. What is a complicit number? A compass. It number is a whole number that can be divided evenly by itself. The number one and at least one other indigent. It has to be at least one other integer. Right, So is nine a conversant number? Okay, let's check a whole number that could be divided evenly amongst the number one. So nine, divide about one. Yes, it works. It can be invited by itself. Nine Divide About nine is one or it works. That'll work for absolutely every single number. And then we need at least one more to decide if nine is complicit or not. So they say nine divided by three is equal to three. At least one other. It means yes. Nine is a composite number. So is seven. A composite number writes a seven divided by one. Yeah, it takes this one we divided by one. But now we need to find another one. It's because we know that one divide of seven divided by seven would give us one. So we know there's needs to We need at least one more. We can only look for indigenous. So just to save invited back to 3.5 No, that won't work. Uh, seven divided by three or 2.3 That won't work. Seven divided by 41.75 So gonna work seven divided by 5114 That's not gonna work either. Seven Divided by 61.17 Knocking the work and seven divided by seven. We know is divided evenly by itself and one but it's not one other Inter Joe. Unfortunately, then we can say no. Those are not indigenous is so No. Seven is not a composite number, right? So can we work out? Is 12 a composite number So we know that 12 divided by one is equal to 12. Right? That's good. Poll numbers 12 divided by 12 is equal to one. That's good. We don't even need to check those. But now we need to try another one. We can quickly see that 12 divided by two is equal to six, which means that six times two is equally 12. So this says yes, it is. A composite number is five a composite number? Okay, lets say five divided by one. We know it works as five. It's good. Five divided by five. Yeah, we know it works. That's good. Canada dry five Divided by two is going to give us 2.5. That's not good. Five. Divided by three is going to give us one. Remained a to or one and 2/3. Uh, that's not good. Five. Divided by full is going to give us one remained, one said Vastly gonna work either. So five mustn't be a composite number. So five is no, it is not a constant number, and we'll figure out soon that five is a prime number. While is 2,343,526. It's a compass of number. We're gonna have to divide it by a huge amount of numbers to figure out if it is a composite number or not. But we've got a trick. If it is an even number, it is always a compass. It number right, so any even number is always a conference. Another. Why? Because we can divide that number by two and get ourselves an answer. So it means is at least another divisible that is not one or the number itself. So this is yes, it is a composite number. His 129 of clubs of number careless try 129 divided by 26.4. No, that's not good enough. Let's try 1 29 divided by three hours. 43. That works perfectly. So, yes, 129 is a composite number. Every natural number is either complicit number or prime number, excluding one. We're gonna talk about one shortly or in the next video. So it's a natural number. So what is the natural number? That's a number from one all the way up to infinity. No negatives. It has to either be a constant number of prime number rise hoops. So conversant number is the opposite to a prime number. If we say seven is not a composite number were saying the same as seven is a prime number. So if we look at these numbers right to these unnatural numbers, excluding one, so we look at 2345678 We know that seven is a prime number. Um, what else can we say? Compass? It means it composes off smaller parts, for example. We know that eight. We know that two times four is equal. 88 must be a composite. Number six is two times 34 is two times two. So these must be conversant numbers. But these numbers only divide by one and itself. So they must be prime numbers because they do not compose off other, smaller parts. Well done. That is our video and composite numbers. Now, we're gonna move on to prime numbers to really understand this for them. 9. Whole Numbers (9 of 11) Prime Numbers and Prime Factors: Okay, now we're going to get into the fascinating world of prime numbers, and prime factors were still only dealing in whole numbers and in number, operations and relationships, Right? So what is a prime number? A prime number can only be divided by one and itself. So you remember the opposite of a prime number is a complicit number, and a composite number can be divided by one itself and something else. So a prime number has exactly two unique positive devises one and itself. This is another definition off a prime number. So is nine a prime number? We did this in the previous video. Proving a number is not prime is easy. It's easy to prove whether or numbers prime allots. It's. It's easy to prove a number is not a prime, but it's quite difficult to prove whether a number is a prime. You'll see that people spend their lives proving with the numbers of prime or not two proven on this prime. You simply need to find one factor besides one and the number that it is devise a bill by so for 99 divided by three that's so easy. No. Nine is not a prime number. What about seven? Now we have to go through like we did in a previous video. Every single interject of visible by 77 times one that works and seven divided by seven is one and networks. So that is a number by itself by divided by one and itself. But if we take seven vital bacteria 34567 we see that it doesn't give us an inter job. So because those are not images, then we can say yes. Seven is a prime number. It is only divided by one and itself. So prime is opposite to a composite number, A positive. Inter Joe is either prime number or composite number. It must be one of the two. So how do we find a pro number when we find that by proving it is not a composite number here? Some rules to help you If a number is even. It is a composite number, i e. If the numbers even it is not a prime number so immediately. If the number ends in a 0246 or eight, we can very easily toll it is not a prime number. This can always be hard. And then there is a factor off to, so it can't be a prime number if a number ends in five. It is a composite number. It is not a prime. Be careful besides the number five. So the number five is a prime is a prime number, so five is a prime number. So don't forget that five is equal to prime. But any other number that ends in a 5 25 1 to 4 to 375 can be divisible by five. And those make those not prime numbers. But five is a prime. A prime number always were in in a 137 or nine. Okay, does the numbers that are left over, of course, besides the number five, the five number rule? How do you practice this? You've got to know your timetables. And it's also good to know the 1st 20 off by heart of your wounds, and we'll show you those. So is number one of prime number from the definition of prime numbers can be divided by one and itself. It is a problem, but hey, but there's another definition for prime. Um, it has to have exactly two unique positive devises one and itself. So one is divided by itself. One divided by itself is equal to one and one divided by one is equal to one s O. But no, they're not unique. They have to be unique. So one is not a prime number. So one is not a prime number. Does not forget that is five a prime number. We just said yes. Five is a prime number. We can divide five by four like we did by 35 Divided by four or five divided back to it doesn't work. Is 505 a prime number? Yes, that's the visible by five. Put that in a calculator. You'll get 101 and you'll see that it works. It's 77,772 upon them. Oh, my goodness. How are we gonna worked it out? Yes, it is because it's even a day. And so that's yes. Oh, that's no, it's not a problem, but sorry. It's 777,000 is 77,772 problem? No, it's even. It can be. It can be hard to copy. A prime number is 57 a prime number. Now it gets a little bit more difficult. Is 71 applying number? He's away. It gets a bit tough. So what do we say? If we said 57 divided by 19 you'll find that that is actually always said 50. So if we said 53 57 divided by two, that's not gonna work. It's not even 57. Divided by three. Ah, that is equal to 19. That works well. So 57 is not a prime number. No, it is not. How about 71? Well, the way we work this out bigger 71 divided by two. No, that's not gonna work. 71 divided by three. No, that's not gonna work on. We're going to do this all the way. I was gonna take forever. And that's actually how you have to figure out. It's the numbers, primal. Not on when people write these computer programs to figure out if a number is prime, it has to go through that order. Right? But here, the 1st 20 prime numbers that you can memorize a few ones is always a fun thing to know. And these air the 1st 20 prime numbers that none of these numbers in this array can be divided by any other number besides itself, it is difficult to prove big numbers. People spend years looking for it. But 71 is your 20th prime number. So yes, 71 is a prime number. So what prime factors? Firstly, what is effective facto are the numbers. You multiply together to get another number. So you would have seen this if you watched the fact his video the factors of eight or eight times one and four times to. So we know that the factors of eight are 124 and eight. Right? But now we only want the prime factors. Okay, so the prime factors of AIDS Oh foreign Teoh, We climbed force, not prime number. But we can divide full by two because four is the same as two times two times two and that is equal to eight. And this is the same as two to the power three is equal to eight. We haven't gone through exponential exponential yet, but you need to know that two times two times two is the same as two to the power of 32 to the power of three means two times to three times. Okay, so what did the prime factors of 12? We know that we can divide 12 3 times four is equal to two on 12. The four is not a prime factor. So we can do this again. We could say three times two times two is equal to 12. Brilliant. And this is could be the same as three times two to the power of two is equal to 12. So one of the prime factors of 147 0 my. So we're gonna have to first figure out what is some factors off 147. So it's first find a factor. 1 47 divided by two is 73.5. That's not a factor. That's not gonna work. 1 47 divided by three is 49 0 that could work a case. And we know that that's a prime factor. So this let's wait for this. But 49 is not a prime factor if you look back, it wasn't on our list off 20 prime factors having prime numbers. But what is 47 divided by seven that is equal to seven. That's quite convenience. So we know that seven times seven is equal to 49. So sorry, what is 49 divided by seven. So we can say that three times 49 is 147 and that is the same as three times, seven times seven. If we wanted, we even put a bracket here. It doesn't matter. And that is equal to 47. And that is the same as three times. Seven to the power of two equals 147. Well done, guys. That is prime factor or prime numbers and the factors of prime numbers. Cool. Now we're gonna go to highest common factors and lowest common multiples, and that is the last section in our whole number section. 10. Whole Numbers (10 of 11) HCF of numbers: OK, guys. Now we in the last section off whole numbers. So in whole numbers, we have gone through all of these various things. And now we're gonna look at high school factors and lowest common multiples. You must watch these two videos if you haven't yet to get a full grass. What effect? And, multi boys, if you're battling, otherwise we're gonna go straight to heis common factors and Lois Colon multiples. So, the highest common factor we're gonna start with that the highest common factor. H See if, as you might see, HC, if is the highest common factor, HC, if between two or more numbers. Okay, so what does that mean? What is effective? The factors are the numbers. You multiply together to get another number. So one of the factors of 12 1 times 12 is 12. Two times six is 12 3 times four is 12. So what are the factors of 12? 12346 and 12. Those are the factors of 12. Now, what if I ask you one of the factors of 15? We know that one times 15 is 50. We know that three times five is 15 so we know that the factors off 15 or 135 and 15 Sonali Sasa question. What is the highest common factor between 12 and 13? Okay, Firstly, what is a common factor? It would be a factor that both these numbers share, So let's look at them. They're both have one great one says cross out ones. Do they both have to know they don't have to. They're both of three. So let's cross out the threes. Do they have a 46 or 12 notes? These are the common factors, and that was something you want to know. What is the highest common factor? And that's very easy. Three. So the answer is the highest common factor between 12 and 13 is three. Okay, let's do some examples of highest common factor. What is the highest common factor between 15 and 40. So, like we just did the factors of 15 or one. It is not too. It is three. It is five and it is 15. Okay, so those are common factors, but those are factors of 15. It's off 15. These are factors. What are factors of 40? Okay, let's have a think what is 40 divided by two 40. Divided by two is equal to 20. So we know that this two and 20 So there's one. There's a to what is 40 divided by three. That as we're fully divided by four, is he going to 10? Okay, so we know that there's four on this side. We know that there's a 14. We know that there's a 20. We know that there's a 10. What about 40 Divided by five that it was 80 so eight and five there's a five and at eight , and those are the only ones we can get. So these are affected of 15. These are factors of 40. What are our common factors? One is a common factor. Three. And to know four No. 55 is a common factor, and 15 is their 15 year No. So none of those. So the common factors between 15 and four a one and five. But what is the highest common factor that is equal to five? And that is our answer. So what is the highest common factor between 18 27 and 63? Okay, so what are the factors of 18? Let's try that the factors of 18 are equal to one, two. And nine said to Let's go make it yet. So there's no we know there's 18. We know this nine. Ah, is there three or there is? There's a three and a six is therefore is there five? No. So it's just that What about 27 rights? What is the factors of 27? We know there's a one we know there's an 18. I'm 27. Sorry. 27. Is there to know it's not even? Is there three? That could be a 33 and nine. Now that could work is therefore no. Is there five? No. Is there six? No. Seven or eight? No. So it must be just 139 27. And what about 63? Okay. What are the common factors of 63? We know there's a one and we know this is 63 right? Is there to know? Is there three All these 33 divided by 63 is equal to 21. Ah, OK, three and 21. That was quite nicely. What about a 45 Niner six? No, that would be 66 or 60. What about 70 We know that seven times tennis 70. So seven times nine Must be 63 are. So there's a seven and a known. Okay, so these are the factors off each of these numbers. What are the common factors? Right. So the common factors are one. We could see this. A one? Uh, no to Yeah. Well, there's a three. There's a three. So the contract is a three. That's good. A six knows another six and nine. Who? There's nines. A cat is a nine, 18 27 and 21. Treason, though, is just that. So what is the highest common factor between these three numbers? The answer is known. 11. Whole Numbers (11 of 11) LCM of numbers: OK, guys. Now we in the last section off whole numbers I can now let's move on to lowest common multiples. So what is a lowest common multiple allows common. Multiple is the lowest common multiple between two or more numbers. So let's start off. What is a multiple? A multiple are the numbers we get after multiplying a number by an indigent. If you're battling with multiples, make sure you watch the multiples video. Otherwise, just remember this definition. So if I ask you one of the multiples of 12. One times. 12 2 times 12 3 times 12 4 times 12 5 times 12 These numbers 12 24 36 48 60 are multiples of 12. It could just keep on going forever. We could just keep on increasing that inter job. So some multiples of 12 of that, including it that way and including it that way, remember, can be negatives. So what are the multiples of 15? You know, there's one times 15 to time served in three times four times within five points of view on we get on multiples. 15 30 45 60 75 right And again that increases on forever. And these are some five multiples of 15. Now, you might need to find more in your questions, but we're going to use these for low. So what is now the lowest common multiple between 12 and 15 right? What if the first the common multiple. So we see that this 12 12 note 12 is note. Uh, 24. No, no. 24. 36. Night is No. 36 48. No, there's no 48 60 or is a 60 year Okay, so that is our lowest common multiple. Eso this. So this would be the lowest common multiple between 12 and 15 and the onset would be 60. We could double that. We could double that to get another common multiple, but it would not be the lowest. It's important looking for the lowest. Another thing I've got to get for mention is you can't use negatives for this. Otherwise, they were going forever. We're looking for the lowest positive common multiple, right? So looking at lowest common multiples, let's try these examples. What is the lowest common multiple between five and seven, right? So what we do, we say five times one five times two times three times four times five is to fight. To start with that is equal to five that is equal to 10. That is equal to 15. That is equal to 20 that is equal to 25. OK, let's do 77 times. One is equal to five and is equal to seven times two times three times four times five 14. That is equal to seven times three is equal to 21 that is equal to 20 AIDS. That is equal to 35. We're not finding one yet. A but to 35 is divisible by five. I know that. So what if we wait six times? Five is 30. That's what they either We're not seven times five, that is five or five times seven secrets, 35 are so this year, 35 is a common multiple and is the lowest common multiple between five and seven. So what is the lowest common multiple between 16 and 15 right? Let's start off with out Big 15 15. We know the 1st 1 is 15. We know the next multiple is 30 can six divide into 38. Conta vied into 15 10 con divide and 15 Ah, the 30 could work. See what is 30 divided by 10 that is equal to an integer off. Three are good said 10 works, 15 works. And what is a city divided by six? That he was five. It's not 15. We couldn't work. So 30 is the lowest common multiple off these three numbers. City would be your answer. Well done. That is Lois comment or highest common factor and lowest common multiples done. Next, we're going to be moving on to exponents. We've done with whole numbers now. Well done, guys. 12. Exponents (1 of 9) Representing numbers in exponential form: parking. Mathematicians were now in the exponents section. The 1st 1 we're going to talk about in this video is exponential form. And what does it mean? Remember, we're still in numbers, operations and relationships, and you can see the rest of videos we're gonna have in the section here. Right. So what is exponential form? Exponential form of an algebraic expression is any expression that includes an experiment. Right. So what is an experiment? Two to the power 33 is the exponents, Right? So in remain. We read this too. To the power of three. That is how are we going to read this? And number three here is our experiment. As you can see, it's smaller and it's at the top, right of the number. So this is exponential form. Other number. Okay, here we have the base, which we call to and exposing three. So number two here is the base, and number three is the exponents in this example. So now we're gonna do an experiment. We're gonna expand it. And what does it equal? So what is four to the power off to? Remember, we say four to the path to Well, what This means is four times itself twice. So that means four times full on what is four times for equal. We know that four times four equals 16. Okay, how about this one? How do we say this? Three to the power of three. What is this equal? This equals three multiplied by three, multiplied by three. And what is this equal? We know that three times three is nine. Said this times this is nine times another three we know is 27. Okay, What about seven to the power off to? And we know that that is equal to seven times seven. And what is seven times seven equal? 49. Okay, what about two to the power of seven? Like so, how do we expanders? This is too times itself. Seven times. So we go two times. Two times, Two times, Two times, Two times, two times two. So that is a cool times. 12345656 and seven. And that is a cool seven times. And what is two times two? We know that that is full. What is four times two? We know that is equal to AIDS. What is eight times two. We know that is equal to 16. What is 16 times too? We know that is equal to 32. What is 32 times two? We know that is equal to 64. On what? It 64 times two. We know that is equal to 128th. Okay, So you're getting the hang of this. Maybe try your own, make up some numbers, say it out loud, expanded, and then figure out the answer. Chicken with a calculator to see if you got it right. So conferred the following into exponential form. We're gonna go through some more examples. What is eight eight is the equivalent off. Two times. Two times two, because two times two is four times to eight and then is to to the path three. What about 16? Okay, so 16 is equal to two times. Two times, two times two, and that is equal to two to the power of four. OK, what about 64? Okay, so what's half of 64? We know that's 32. And what's half of 32? We know that 16. So we know that if we said two to the power of four times, another two times. Another two, we would get 64. And that is the same as two to the power of six. What was two to the power of seven? Remember, we just worked that out in the previous slide, and that was 128. And that is simply times in 64 by two. What about one? How can we convert one to exponential form? Well, the beauty about one is we could do this. We could say one to the power off one. And that is equal to one. What about one to the power of 27 that is equal to one times one times one all the way 27 times. And that is still unequal. One. Another thing we could do is we could say what is four to the power of zero. What is four to the power of zero? I'm not totally that rule right now for to the power of zero is equal to one. 27 to the power of zero is equal to one. Absolutely any number X to the power zero is equal to one. And we're gonna learn about that rule in the next video. But all you need to know is any number is raised to the zeroth. Power is equal to one. What about 27 while we dealt with 27? If we try, divide 27 by two readers 13.5 and that doesn't work. So if we divided by three, we get nine. So if we could go, that is equal to three times nine, which is good. This is getting us close. But nine is the same as three times three. So that is equal to three times three times three. And that is the same as three. To the power of three. I can. Okay, keeping with exponential form. Let's expand these so five to the power of to we say five squid and we're gonna deal with later why we call this a squared. Especially when you look at square roots, right? But the reason we do this five squared is equal to 25. That is the same as five multiplied by five. Right. If we want to define the area of a square that is five wide and 55 high and five wide the area of the square is equal to 25. It's five times five, and that's why we say this is five squared. What about four to the power of three? We say this is four cubes. So what is for cute? That is equal to full times full times full? What is four times for? We know that it's 16. What? It's 16 times for money to get a calculator or do your long multiplication for this. Remember, you can always say 16 times. Four. Put that under what is four times six. That's 24 and then we add up to what is four times one is four plus 26 and that is 64. That is what you can always do with exponential form if you're battling. And the reason we say this is four cubed is because if we have a cube, if we have a cube off length for and hide fall and with for so that's 44 and height fall, the area of this cube is equal to 64 units to the power of three cubed, so that is what we say. This is four cube, so a two because squared at of three record cute What is this to the par zero? Remember? Absolutely. Any number to the power of zero is always gonna equal one. We're gonna show that No, What is 151 to the power of one? That is that times it's nothing. 151 that's a 506. So any number to the power of one is its number itself. What is 10 squared? 10 squared is 10 times 10 and that is equal to 100. I can't. What is 10? What is 10 to the part of six. So that is 10 times. 10 times. 10 times 10 to 10 times. 10 times, 10 times, 10 times, 10 times 10. And that is equal to one million with six zeroes rise. So why do we use exponential form? So it's a great way to show big, big numbers. It is much easier to write 10 to the power of six than it is to write one million. Okay. For example, you could say 10 to the power of 120 and this is much easier than writing 120 zeros. So this I exponential form at the end of each video. I'm a show you those charts for a couple of seconds and you can look at this. Don't stress too much about all of this over here yet. I'm just gonna show this as a reference. It's a cheat sheet or a way to remember the rules. And by the end of this video section, I want you to remember this entire sheet. But all we learned now was exponents is a base and an experiment. We next to the going to scientific notation. So write these down if you need, but they will be at the end of every video. 13. Exponents (2 of 9) Representing numbers in scientific notation: can. Now we're gonna be doing scientific notation and you'll see white in the exponents section , right? So scientific notation is a way for people to handle very, very large or small numbers. Right? Let's have a look. An example to better understand five times 10 to the power of negative 10 or 150 times 10 to the power of nine. Right, So this is an A a scientific notation. So the notation is the way it looks. That's what notation means. How it looks and scientific is because scientists uses a lot when they dealing with very small things are very large things. For example, if you see and to the power of negative, it is a really small number, it's very, very small. So when you see a negative, it's a very small number on when you see a positive, it's a really big number. It's OK, So what I want to show you here is, why would we use it? This number is exactly the same as this. So five times 10 to the power of negative 10 is exactly the same. A 0.0 is a reserve along the way So this number is exactly the same. Except we've moved us. Come over here. 123456789 10 times. And that's why we get negative 10. The way you know this is correct. That you go in the right direction is because you can see how tiny this number is. What about a raid? Big number instead of writing 150 1000 million rights or trillion, we're going to just write 100 50 times 10 to the power of nine. What we've done here is you take the decimal that used to be here. We moved at 123456789 points across. And we got that day. I'm gonna show you there's no. But for some import, this in meters is the size of hurtem, or this in meters is the distance from Earth to the sun. So is much easier to use scientific notation. And that's why scientists uses these are very scientific things to measure. So when do we use scientific notation? 4.7 to 89 times 10 to the power of nine. This is perfect. Remember, we have to always include these numbers. It is not zero. We've got it included. Another time is 320. Would you really want to go 3.2 times 10 to the power of to No, it's actually easier to just write 320. So we only deal with it when we doing really very big, large or small numbers. Okay, let's have a look at how to do it. So we got 532,000. This is the same as 532,000 0.0, right, So we can all move those point across. So we've moved this 0.12345 We've moved it five to the left rise. And this is a very big number number that we're making smaller. Okay, so this is showing that we've moved at 12345 across, and because it's a very big number is going to be positive times 10 to the power of five. Right? So there's a 12345 across, and that is the same as 5.32 times 10 to the power five. We don't have to write those heroes, but we have to write the three and two. What about this number show? There's massive number here. How are we gonna work this out? Okay, so this is the same as that to the power zero. So we're gonna write five 0.5 65 seven Rights, 5.5657 Excuse the handwriting. And how many times were going across? 123456789 So that is times 10 to the power of nine. Because it's a really big number. What about this one here? 0.0 17. Right. So we're gonna right? Seven 0.0. How many times were going across? 12345 times 10 to the power of negative five. Because that's a really, really small number. Okay, let's try some more examples. What about this big number here? Right, So we're gonna have to go 67006 rise. Now, we have to include those two zeros because we need to use a six times 10. Right? So this is where our point is, so removing it from over here today and that's 1234 across. And it's a really big numbers of 10. Sorry. Times 10 to the power off four. Okay. What about this number? Sure. So, Yeah, we can simply go five points year old times 10. There's a really big numbers is gonna be a positive. And that is how many times a going across here we are the same as 0.0. So go. 123456789 10 11. So this is 5.0 times 10 to the power of 11. What about this member again? We're gonna have to say 5.4 0007 We need Include that, seven. Now, we're gonna move this 1234 points, and it's a very small number, so it's gonna be 10 to the power of negative four. How do you convert this back? Okay, so we're gonna need to move this. This is a big number, right? Is positive. So we're gonna need to move those three to the right. 123 That's convenient. So it's 6000 and seven, and that is our own suit, right? That is all we do. That is the exact same. In this case, it might be easier to just show it like that. Whatever. The about 0.5 times 10 to the part Negative for Right, So we need to go. This is really small number. We need to go that way because it's really small. So we're gonna add some zeros in year 0000 and we need to go for a cross. So we kind of here, we're gonna go. 1234 Right. So this is zero point 12344 zeros. 12345 Okay, what? About 10 times 10 to the power of 100. We not going to write the salves, But this is gonna be 10 with 100 zeros behind it to get an idea of how this works. So we've not done signed two fig notation and you can take some time to have a look at this . Remember gonna show this at the end of each video just so you can get a better idea. It's a sheet to help you remember at the end of the section. And this is how we did scientific notation with positive being really big and negative being really small. And remember, we've got to move the decimal in different directions. Okay, Great 14. Exponents (3 of 9) Power of Zero Law: Okay, Now we're gonna talk about the power of zero law, rightly stolen exponents section and the next couple of videos. We're going to deal with a whole bunch of different laws, but the 1st 1 is immune to have a zero power or zero law. Right? So let's start off with the law and I'm gonna give you an example. We're gonna first do what is X to the power of one, and that is equal to X. An example of this is full times. Nothing for just by itself to the power of one is equal for right. So for any number, it's to the power of one equals X. So writing that is exactly the same. So if I said 270 to the part of one, it is the same. Is writing 270 right? What is X to the power of note and that is equal to one. So any number X to the power of note is equal to one. Remember when I write X here, I mean any number for example, six to the power of note equals one. What is 270 to the power of note that is also equal to one. Any number to the power of zero is equal to one. What is X to the power of negative one, right, Excel apart. Negative one is the same as one over X. Okay, so any number X to the power of negative one is the same as one over X. It's an example of this is three to the power of negative one, and that is the same as 1/3. Okay, so if I said 270 to the power of negative one, it's the same. The same one Over 217. Okay, what about five to the power zero That is one and is equal to one. What about five to the power of one that is equal to one times five and that is equal to five. Okay, what about five to the power to that equals one times five times five. And you know that equals five times five times one is 25. What about five to the negative one five to the negative. One is the same as one divided by five. And that is the same. A 0.2 right so if you could 1/5 into your calculator, you'll get 0.2. What? I five to the public. Negative to. This is the same as one divided about five divided by five. And that is the same as zero. Going zero for five. And we can write that as 1/5 squared. Okay, Don't get took of yours by the two divisible signs there. Okay, let's just have a look at some examples. What is X to the power of note? We know that is one. What is 12,546 to the power of zero? That is one. What is 1/2 to the power of zero have drawn is quite badly. Doesn't mean to be brackets here, and that is equal to one. What is the number pi to the power zero. And that again is equal to one. What is 1/2 to the power of one that is equal to 1/2? What is pi to the power of one and that is equal to pi? You might look not learned by pious, but you definitely will in the future. So don't worry about those two. If you haven't heard about it yet. Let's just have a look at a proof you you're gonna learn later. A law that says X to the path M divided by X to the power in is equal to X m minus in This is cordial zero and your division Law on. We can use this law to prove that H to the power of zero is equal to one. So it's to the path. Zero is exactly the same extra power four divided by its the far four and that is equal to one. And one is the same as that which is equal to four minus four using this law that, minus dad's so four minus fall right and that is X to the power zero so x to the power zero . Forgetting about even this first section X of a fall of except for is equal to one, right, So it's a force except for is equal to four minus four, and that is equals zero. And that's why X to the power zero is equal to one. So you oops, you only need to worry about this section here, but don't worry about that. This is just proving why aced the par zero is equal to one. But all you need to know is that any number to the power of zero is always equal to one. So that's a part of zero long. Now I'm leaving this year for you every time to read over as a reference. And I'm believers this open for a few seconds. Write it down. By the end of this video section, you would have remembered all of these equations. We've done exponents and scientific notation. Now we doing zero and the one law, right? So we remember X to the power of one is equal to eggs extra part zero z could one x to the power of negative one is equal to one over x great. 15. Exponents (4 of 9) Multiplication Law: Allah mathematicians. Today we're gonna be doing the multiplication law. There's a still in our section of exponents and like I said, we currently dealing with a couple of the laws, right? So today is the multiplication of exponents law. So let's start by stating the law The law is X to the path in times X to the path in equals X m plus in right the finger Nita, no year is in and in our difference, that can be the same. X and X are the exact same. So x and X have to be the same, right? So let's look at this x to the power three times X to the powerful, right? So this is the same as three X is 123 The dot simply means times Times four X is which is 1234 And you know that that is I've just spread it out for you. If you've never seen the dark before this what it means, it means at times and that's the same x to the power of seven, which you can see is the same extra the three plus four. So we've gone from this over here to that from that day to that over there. Okay, let's have a look at some examples of this. What is X to the 12 times X to the full right. So that is equal two X to the 16. Right? We just add those two together. What about X To the hundreds, Times X to the 34. Again, this is just going to simply be X to the 134. Okay, we just adding those together 100 plus 34 because this is x zero X must be the same. Remember that I'm saying, if it was two squared times too cute, that is exactly the same as saying to the power of five. But because thes twos of the same as in the Xers in wouldn't work. If you said two squared times three cute. You couldn't simply say this is equal to that times. That which is six, for example, And then that cause that and that does not work here. Okay, so it has to be X. It has to be the same. Let's have a look at the law again. And some more examples. What is two to the power of two times two to the power fall. Right, That is equal to two to the power of six. Like we've just done to us too. Make sure you don't multiply. That's a plus that is equal to two. Two plus four. Okay, what about two to the power 3 to 2 to the power of three. Right? Unfortunately, Conta, anything in this case would have to go to to the path three multiplied and putting bracketing to make look easier by three to the path to what is two to the power three. We know that that is eight. So we haven't ate there. And what is three to the path to? We know that is nine. Because there's a bracket yet, which is the same as a multiplication. This is the same. A 72 right? Let's have a look at another example. What is 10 to the path to times 10 to the power to. And here we can write that is the same as 10 to the power of full. Okay, because they brought the same. What about X squared plus x 1/2. Okay, so this is the same as eggs to plus 1/2 right. So how do we go to pass 1/2? We know that too Cassa have is the same as for over two. Because ah have which is the same as 5/2. Remember, we have to get a lowest common denominator, and that is to And that's what we did there. Because for over two is the same was to, And now they bite the same weaken. Say, five. You too. So that is equal to X 5/2. Right, Let's try this one x three of a full times X to the power of 1/2. So that's eggs. Three or four plus 1/2. What's three of a four plus 1/2. Right. So what is that equal? We're gonna say three of a full plus 1/2. We need a lowest common denominator going to see that easy. That that is full 3/4 plus two over form. And that is the same as 5/4 rise. You could if you wanted, right. One and 1/4. But we gonna leave It is this and that is equal to X to the power of 5/4. Great. So that's your answer Let's have a look at the law again. And some more examples. What about X To the power of three times X to the negative one. What about extra part three times X to the negative To how are we gonna do this, right? We can just minus it. But this is going to get us into our divisional, right? So we can say that that is equal to x three plus a negative too, which is the same as X to the part of one which is the same eggs. But we're gonna rather show you the divisional now. Okay, so that's a multiplication law. And I'm gonna show you this at the end of each of your videos on the sisters to get you updated, you could pause it here and write the equations down. We've already talked about exponents, scientific notation zero and one law. And now we've just done a multiplication law where it's to the M times Extra then is equal to X m has in important thing is these are the same number x and X. It has to be the same Great 16. Exponents (5 of 9) Division Law: OK, guys, Now we're gonna be doing the division law, right? We're still in the section on exponents and we now going through the laws. We have it first. Did the powers you're a law in? We did the modification or we now doing a divisional before we go into the power law. So the divisional, let's start by stating the law we have X to the power in divided by X to the power of in on . This is equal to X m minus in Ryan's. So it is important here in can equal in, but it doesn't always have to, but the exes have to be the same, right? So let's go through an example x to the power of five divided by X to the power of three. So that is X times X times X times X times X divided by X Times x times X The dot simply means times it's expanded salves. We've put the extent of the five x on the top of the three x on the bottom, and we know that they just cancel each other out because they're like tunes, Right? Canceled three on the top in three on the bottom. So we left with one on the bottom and this is the same as X to the power of to which is the same as five minus three. So this is showing that that law goes from this year Extra five divide based in 32 x five Ministry. This is really just an extension of the divisional. Another multiplication. All. Except now we have a negative number. Let's have a look at some examples. What is X to the 12 divided by X to the fall. We know that that is equal to eggs. 12. Mine is full, which is equal to eggs to the eight rise. But like I was saying, this whole thing could be written like h to the 12 times X to the negative full. And then we simply have an extension of our multiplication rule where we go next to the 12 plus negative full and that is equal to X to the eight. Exactly the same. So divisional is simply an extension oven multiplication all. What about X To the 100 divided by X to the 34. Right, So this is equal to X 100 minus 34 and that is the same as aches to the power of 66. Okay, let's have a look at some more examples. So we have to to the power of to divide about two to the power of four. Right? So this is the same as to to the power of to minus four, which is the same as two to the power of negative, too. What is two to the power of negative two? We know that is the same as 1/2 squared, which is the same as one of a fall, which is equal to 0.25 But that should be fine. Even that would be a can. What about two to the power of three, divided by three to the power of two or two cubed, divided by three squared. Because these are not to say we can't. Unfortunately, what? Do anything with the powers here? Because that X and that X, these are not the same. There's a tour of here in a three of year and two does not equal three. So here we have to know what is two to the power three that is eight and what is street to the park to that is known. And that is our answer. 8/9. What about 10 to the power of three? Divided by 10 to part two, that is simply equal. 10 3 miners to which is equal 10 to the power of one. And that is the same as 10. What about X squared? Divided by X two a half. Okay, that is the same as eggs Tu minus 1/2. Right. And we know that that is equal to what is two minus 1/2 is equal to the set to is the same as for over two minus 1/2 of getting the lowest common denominator. And that is equal to 3/2 so x to the power of 3/2. What about X To the power of three of full divided by X to the power of one. We're gonna do the exact same thing. Really? Set X to the power of 3/4 minus 1/2 right, and that is equal to eggs. What's it to the power? Let's go. 3/4 minus 1/2 is get the lowest common denominator for four miners to over four, and that is equal to one over full. Right. So that is the same as X to the power of one over full. Okay, they try some boys armfuls. What if we really say X to the power three of a fall divided by four to the power of to Now for this one, we can work this out, but you can tell that fool is the same as two squared. So what if we would have changes to to to the power of three the but divided by two squares squared. How are we going to do that? Well, we're gonna use power laws, which is in the next video. So we just turned the divisional on at the end of each These videos. I'm going again show you this where you can pause it and write these equations down. But this is the whole section on exponents. We first learned about a basic exponents. We did some work on scientific notation which previously did the multiplication All this year, a law. And now we've just done the division. Or And we looked How closely linked these two. Ah, Now we're gonna go into the power laws 17. Exponents (6 of 9) Power Law: Okay. Mathematicians now we're gonna be doing power Law. It's the last of our laws in the experiments section. Let's start of that law X to the power of em or to the path in is the same x m times in There's a multiplication, their little doctor time sign. There it's looking Example. Extra the three to the power of three. We know that that is extra. The 3123 three times 123123 And that is the same as X nine times. And that is the same as Extra the nine, which we can see his x three Times Street. So we went from there and we came from there. Write em in in, in this case with the same and that's fine. Let's do some examples. What is X to the power of 12 or to the power of full right? So we know that that is equal to x 12 multiplied by four. What is that equal 12 times for is 48. So this X to the power of 48. What about its to the 100 to the power of salmon? So this is simply X 100 multiplied by seven, which is equal to X to the power of 700. Okay, let's have a look at another power law. What about X? To the Y to the n is equal to X to the n times whiter name Right now, in this case, we have different exes and different wise. But when we square the whole or win, we put raise it to the power the entire bracket, each individual term in their bracket, gets raised to the same power. Let's have a look at an example x times y to the power of four. It's important that you know where your brackets are. That is X times y four times its times why it turns. Why it's times what we dropped the brackets. We move it around This four exercise this four wise and we can simplify that to extra four times y to the full. So we've gone from there on. We've come to here, right? Let's have a look at some examples. What is X times y to the power of half that is simply equal to X to the half times why to the half There's nothing more we can do that What about full times X to the power of three Rise. This is the same as four to the power of three times X to the power of three. What is We can simplify this further if we wanted, although that is okay. But what is four to the power of three when her foursquare to 16 16 sons four is 64 so it's 64 times X to the power of three is your answer? Okay, let's have a look at the last power law. We have X divided by wire to the Parvin, and that is the same extra end divide about why, to him exactly the same as our previous one. But now we have a vision sign. It's work this out. It's divided by wives, the powerful that's X divided by why it's divide about why it's divided by why extra viable wife four times. And that's the same cause, all the exes on the top and all the wise on the bottom. So it looks like this, and that is the same x to the power of four. Divide about white, the powerful. So we've come from Yeah, and we've got to there. Okay, Let's have a look at some examples. What is next to the hearth? Divided by white off that is the same his ex to the whole of divided by Why? To the hearth grace One more four divided by X cute that is the same as four to the part of three. Divided by X cute. And that is the same as we know from a previous slide at 64 divided by X cubed. Great. Let's not try some examples of these laws we have are three laws. Let's try four X trade. Divide them to the power of three. Let's try four times X squared to the power of three. Right, So this is equal to this two terms here the fall and the X squared. So that is full cubed times X two times three. Remember, this is the same as one times three over here. So what is four queued? We know that that is 64 times X to the part of five or six. Because we multiple kind pi ing it together. Okay, What about ex squid? Times y to the power of 1/2. Okay, that is x two multiplied by 1/2 times what a one multiplies by half and that is equal to two times 1/2. We know that to use the same is true for one Those cancel out and we left with ones. So you know that that is X to the part of one times y to the power of 1/2 which is exactly the same x times y to the power of 1/2. Okay, let us try no X squared, divided by wire to 1/2 or to the power of four short. This looks big, but it's really easy. This is the same as X to times for divided by why 1/2 times for do you see how easy that Waas What is two times for we know that is eight But what is Wife's the hard times for what is full times one hot. So what is what is happening if we have for that is equal to y squared? And what about this example now X to the power of two times X to the power of half times y to the pub zero or to the power of 1/2. Okay, what do we know about power to this year. We know that equals one, but I want to show you exactly why it will always be the same anywhere. So what? We know that that's exc Wage Times went off times x to the Hove Times went off times Why to zero times 1/2 What is exe squid times 1/2 That is one as we didn't know previous example What is 1/2 times 1/2 times to talk, We get one. So that's X and we times the bottom when we get two times two is full and then what? Zero tons, 1/2 We know that that is always equal to zero. So that's gonna equals zero. And what is anything to the par 01 like? But we're not finished yet because now we can we timed into so we can go x one plus 1/4 rights and one is the same as for over full. So for over four plus 1/4 is 5/4. So that's exactly the same X times 5/4 And that right there is your answer. Okay, so we just down the power laws. If that was a bit confusing, don't worry we're going to do some examples and operations. And lastly, I just want to show you this cheat at the end of every move on video. I just want to show you visit the end of every video, pause it and write down these laws less. You should not have finish. We've done exponents scientific notation. 01 law, multiplication or divisional. And we've just done the three power laws. So now this you need to remember and practice, we're gonna practice it in some examples, But first, you're gonna do cubed and squared roots. So it pulls your video and write this down. Well done, guys. 18. Exponents (7 of 9) Finding Square Roots: Okay, Mathematicians. Now we're gonna be doing square and cube roots. This is still in our exponents form. We just finished with all the laws. And now we're gonna go onto what is the square and a Q. Bruce. Right? So squares are rectangles with all the sides of equal. You know that A square simply rectangle where all the sides of equal a number is a perfect square. If you can make a perfect square farm, that number of 1.1 unit squares what I mean by that. So this is Ah, square. Where? This with one and height. One. Right. So this is a perfect square off one by one unit. So a number can be made a perfect square if we can make that number Inter perfect. Sway off one by one. Units, Wait. Don't get too confused is for perfect square. So we have 41 by one blocks year. That's one by 11 by 11 by 11 by one. And you know that if we group these one by one into a square like those, it is making a perfect square because we can make a square after those one by one, So we know that's 1111 So what is that equal? One plus one is two. What about one plus one here? That is equal to two. So we know that the width and length and height of the square is two and two. So we're gonna use this symbol here. Rights to show square root. Okay, so this is the side length of the square and number forms is called the square root rights of the side length of a square. Another forms is called the square root. So we have a number and our numbers for so a number. Right? That's full. The silence of the square off a number is called the square root. Right. So the silent off the square. So we made for a square because we had to buy two. There were four ones and made two by two. And then we go the side length and that balance is equal to two. And this is called the square root. Okay, so if I said the square root of fall, that would be equal to two. Okay, you getting this? So the square root means the late off the square, so now here we working with perfect squares. Let's have a look at his five. A perfect square to re five squares. Each of these are one by one. We try to make a square out of this, but record. There's nowhere we can make a square out of these five little squares, so No. Five is not a perfect square. However, the square root of five can still be calculated to make 2.23 on each side. It's not a perfect square, but if we had to turn five into a square, we've got 2.25 times to point to fire at 2.23 times 2.23 and that will give us five. So if you said 2.23 times 2.23 you will get five, right, so that area is equal to five. Cool. What about its six is perfect square. We have six ones. These are one by one rise. We can't make a square. We make rectangles where that length is three and that lenders to or there's two there and three day. But we can't ever make a square, so no, it is not a perfect square, but it's still a square because if we say the square to six, we put it in the calculator. We're gonna get 2.45 and that's the woods of each of these sides here. We're gonna mostly be dealing with perfect squares. But I want to show you that even if it's not a perfect square, it still has a square root. Right. So is nine a perfect square? Yes. Look, you have one by one by one by 111 So that's nine. So yes, it is a perfect square. So what should we say? These lanes, all these lines are each three three every year and three other. They're both equal sorts of square. So the square root of nine is equal to three. Are you getting that is 16 of perfect square, right? So it could 16 blocks. We got a perfect square here. We got four in the top and four long live side. So that yes, it is a perfect square. The length of each side is full. So the square root of 16 is full. This is the same of, say, three squared is equal to nine and four squared is equal to 16 so you can work backwards to do. Your square root is one of our five. What is five squared? Five squared is 25. So what does that mean? The square root hoops. The square root off 25 is equal to five. The square root is almost the opposite to the squared. What about six squared? Yes, that's 36. You know that. So therefore, 56 Quaid, a square root of 36 is six. What about in squared? Hey, Well, we know that the square root of n squared is equal to end. They cancel each other out. Okay, Okay. Let's do some examples. What is 24? The square root of 25. We know that is equal to five. Okay, what is the square root of 144? Right, So this garden do some numbers. Yet we know that 10 times 10 is 100 so called beating. What about 11? Let's try 11. Not supplied by 11. That is equal to 100 and 21. No, that's not rides. Okay, What about 12 is trying 12 multiplied by 12. Because remember, this is the same as 12 squared, and that is equal to 120 past 24. That's 144. Hey. Okay, Remain. We can always do this like this if we need to. If we if we struggling 12 times 12 and do long multiplication or multiplication by columns . So this ours is equal to 12. Brilliant. What about 225 4 How are we gonna work this one else can we know that 12 times 12 is 144? Why don't we try? Multiply? Let's say 16 by 16. Okay, so this is a big number. So let's go like this. 16 multiplied by 16. We're gonna do a long color multiplication. Six times six is 36 plus a three six times. One is six plus three is nine. We add a zero one time sixes six and one times one is one disk unequal, six attained, we can really see is looking equal. Five days on nine past six is ah, nine past six is 15. We have a five and a one and that's 256 0 So this is too big 16. Let's try 15 times 15 So we're gonna go 15 times 15 five times five is five. We add up to five times one is two. We had about seven. We add a zero. One times five is 51 times one is one. We got a five. Seven past five is equal to 12. Got out too. We had a one of the top one. Past one is to a There we have it, fellas, is 15. So you've got to keep doing. There's going above the number like we did there. You might drop below the number, and eventually you'll find the number. What about the square root of 100? Ah, that's easy. The square root of 100 is the same is equal to 10 because we know that 10 times 10 is equal to 100. What is the square root of one square root of one is one. Because you know that one squared is equal to one. What about the square root of 20? Okay, we know that the square roots off 25 is equal to five. Okay? We know that the square root off 16 is equal to fall. There is no number in between. There That's a whole number that can equal 20. So we know that 20 has to be somewhere between 16 and 25. What were going to figure out is that this number equals 4.47 You have to do this on a calculator, but we're not gonna worry too much about this because it's not a perfect square. Okay, we figured that out because there was a five. The squared of five years. 25 the screeners 16 when you was full. So they couldn't be anything in between four and five that could give us a scrape through to 20 for practice. I want you to go and find find end square from n equals one to n equals 20. I want you to go and say one square and see where that equals two square and see where that equals three squared and see where that equals. And I want you to write this down on big piece of paper or the way to 20 squared and see where that equals. This will really help you. So if Donald Square Roots and Cube roots, I'm gonna leave this up for a couple of seconds for you to copy down if you want. This is all the laws we figured out in the previous video 19. Exponents (8 of 9) Finding Cube Roots: OK, guys, now we're gonna be doing the Cube roots. Previously, we had done the square roots, and now we're doing cube roots. We stolen exponents, and it's an important part of exponents. So what do you think about when you always think about a cube? A cube? A Rubik's Cube rise? So Rubik's Cube is a great example. There's one big cube, each made up made up of a whole bunch of smaller cubes. Can you figure out how many smaller cubes are in this Rubik's cubes? Okay, we know this one here. There's nine on each side. So there's nine here and we can say you see that there's three layers of nine. So what's three times 9 27? So ruby execute is a big cube made up of 27 small cubes. Okay, remember that. So what is the perfect cube? A numbers of perfect you. If you can make a perfect cube from the number of one times one times one unit cues, right? So in our River cube we had 27. Let Foy's arbol one times one times one unit cues and then number 27 made up a big, perfect cube. So that number was a perfect cube. 27. So let's have a look. We have one times one times one times on Ennis Acute, such times one. So we have one height, one with and one depth. We have a cube. Okay, so is four perfect cube. So this take four cubes for one by one by one cubes, and we stack them all where we're going to try and make a big cube out of it. But we cons we can make a cube out of Ford's. This is really just a flat square or flat tube is not a cube. Um, so four is not a perfect cube, but what about eight is eight of perfect you? Why don't we add another four cubes to this and we put them right on top, and we can see that we've created a perfect cube where the highest to the witness to and the length is too. You can see that because there's full cubes in the top of one by one by one by one. And full in the bottom of one by one by one by one. So, yes, eight is a perfect cube. And to express a cube root. We use exactly the same as we did for square root, it said. We have to show a three here for cubed three times, and it can be any number underneath you. So this is equal to a cube root the size length of a cube and number forms is called the Cube root, right? So what is the Seidlin off a cube? The number eight forms. So the number eight the number forms the silence of the cube that number forms was, too, because we had eight cubes here and each length was too. So the square root of or the cube root of eight is equal to two. So this is the same as two times two times two times two. That is equal to age. What is this? Also the same is to to the power of three or two, like Rick or cute. Okay, so we think about our Rubik's Cube, How many smaller cubes and a Rubik's Cube? We just decided there's nine on the side. Here's 1234 56789 And there's three layers of this of his nine times. Three this can also be said is three times. Three times three. Okay. And that is equal to 27. Because we had these three. Or rather, those three times. Those three times. Those three. And that gave us 27. So the cube root of 27 physical to three. So what is, for example, three Cute? That is the same as 27. Okay. His 64 perfect perfect cube. I've drawn this for you. We're 64 that is the same as four times four times. For you can see this late. There's four here on the height. The length is for and the death is four as well. And this, if you can't will make 64 cubes. We know that that is this four times this four times this for to get a 64. So the cube root of 64 physical to fall. You're getting the hang of us. Let's try some examples. What's a cube root of 1000? All that seems scary. But don't be straits because 1000 is the same as 10 times 10 or multiply by 10 multiplied by Tim. Sorry. That is unequal. That that is the same as 1000 right? Wants to strike that Syria there. So the cubed root practice writing this cute of 1000. What multiplied three times will get us 1000 10. Okay. What is the cube root of 125? We know that five times five is 25. What about multiplied by another five? And that is going to get us 125. And that officers in simply five. What is the cube root of 27? We've done this. That is equal to nine. Sorry. That is equal to three. Because of three times three times three is equal to 27. Okay, What I want you to do for practice is go from in eight equals one to n equals 10 and find a cube roots. I want you to find what, 812 the three years, right. And then I want you to find what is that equal? What is to to the cube equal What is three to the cube? Equal all the way to 10. If you're feeling really bread, you can keep going up to 20. See if you can find their zones. Onda. After that, you can figure out that the cube root is exactly the same. So if you have to go any number 227 que berated. And then we were Cuba. It the whole thing that is equal to 227 because they cancel each other out. Okay, give that a practice that was in a square cube roots room. We're doing cube roots and exponents section animal. Leave this up for a couple of seconds. Poor civilian. Er, these are just a reminder of our laws. 20. Exponents (9 of 9) Exponential laws four operations examples: Okay, everyone, we're gonna now do some examples of heart youth. All these experiment operations. So all of the laws, all in one. So this is really this video is just gonna go through some examples and talk you through? Have to do it at the beginning of each one. I suggest you pause the video, you try it yourself and then you watch how I walk through it. OK, so let's go. What is 27 to the power of one? This is just reminder that is equal to 27. Remember, any number to the power of one is equal to that number. So that is the law where X to the power of one is equal to X. Okay, what is 16 to the power of zero that is equal to one remained that any number X to the power of zero is equal to one. What is four to the power of negative one? We can't remember that anything to the power of negative one can be the same Is writing 1/4 to the power of one. What is four to the power of one that is simply equal to fall? So that is simply go to one of the four and you can work out on your calculator that that 1/4 is the same drew zero as same as Europe, 1 to 5. What is fall to the power of negative one or to the power of to Okay, now remain. We have brackets here. So what do we do when we got Brett? When we have brackets, we multiply our powers. So that's four to the power of negative one multiplied by two. What is that gonna equal that's gonna equal for to the power of negative too? Okay, we're not classing we multiplying. And now what's four to the power of negative to? That is the same as one over four squared, which is the same as 1/16. And you can work that out on your call Clarity. A few words can What is forced apart? Negative one to the power of negative two or multiplied by 14 Power three. Sure, this looks complicated, but do you know it's trace? Let's first start with this bracket. Four to the part. Negative one or two. The pardoning end of two. It's exactly the same as those. But remember when It's a power. We have to multiply it. So that's gonna be four to the power of negative one multiplied by negative too, all in the brackets. And that is going to be four to the power of three can. So what is negative? One times negative too. We know that when we a negative times a negative, you get a positive. So that is full squared, right? Negative. One times negative two is equal to two. And that is all multiplied by four to the Cube. Can we have the same base here? So we can add these together because we multiplying the base. So that is the same as four, two, plus three. And that is equal to four to the power of five. And that is fine. If you want, you can calculate what for? To the power of fivers. It'll be good practice. I can't. What is seven squared plus seven to the power of one can. What do we need to be careful about here? Can we go seven squared two plus one or 72 plus one? No record, because that's a plus is not of multiplication. So for this example, we actually the best way to do is you could leave it like that. Or you could go what seven squared. And that's 49. And what a seven. That's plus seven. So what's 49? Past 7 56? And that is your answer. Be careful. You conquer 72 plus one. That is not right, because there's a multiple. If it was seven squared times seven to the part of one, then you could do that. But you call it because it's not like that. What about five cubed minus 53? Exactly the same thing here, So we're gonna have to go. Five cubed is 125 minus five squared, which is 25. And that is equal to 100. Okay, Same example. I want to show you. Now what is forced apart? Negative. Two minus four to the power of negative three. You've gotta go 1/4 squared, minus 1/4. Cute, right. And then you can work out with these Are So what's four squared? 16 says well over 16 minus. What's for cute 64. Whenever 64 we need to find a common denominator here on. We know there's gonna be 64 and four times 64. We're gonna add Afford a top. So there's 4/64. That's the same is one of a 16 minus 1/64 and that is gonna equal 3/64 were carried. This is just a practice with your fractions. What is full times X to the power of negative one All squared. Okay, so this two terms in the other four turn and the X to the part. Negative one and it's all being squared. What do we do? T power? We multiply it. So that's full one multiplied by two times X negative one multiplied by two. Right, So that is gonna equal for one times two's, too. So that's four squared times X, Nate and negative. One times two is negative. Two X negative, too. Okay, what is foursquare? We just done that that 16 4 times for and that X squared is times one over X to the neg or two squared, and that is exactly the same as saying 16 over X squared can. So that's that could be 16/1 day. Can What is two squared at times X to the power of negative to all squared. OK, you'll find that two squared is exactly the same school, and it's just another way of showing it. So what are we gonna do yet? We're going to say to two times two, because it's to the power times the eggs minus two multiplied by two. Okay, so what is two times? Two to the power of force that is equal to two to the power of fall multiplied by X to the power of negative full. Okay, what is two to the power of full? Can you work that out? That is the same. A 16. Over. What? We're just gonna right to keep it. The Sanders, that's and what is X to the negative. Full. That is one of the X full, which is the same. A 16 over x four. Okay, you getting this? What about this example? Two to the power of negative one over, divided by four. Or to the power of to. So we just simply multiply out powers because there's to the power. It's not a time, so we multiply for squared. So now we have what is to depart negative. One times two and four squared that these up showing is over one in one tomb. I'm no just split these up. It's still want to him, but I'm right to to the part of negative one, which negative one times two is negative. Two. That's gonna put it over one. And I must say that is times 1/4 squared rise, which is what we have here. What is two to the power of negative two? That's the same as 1/2 squared that's one over two squared, multiplied by one over fourth grade. What's 1/4 squared? Whatever force greatest 16. So this is the same as one over full multiplied by 1/16. And what is that equal? Two. We multiply the top that we get 11 times. One, we multiply the bottom four times. 16 is 64 that is on. So what about this question? Very similar to the one of the above. Let's go two to the power of negative two multiplied by three all over one, and that's one that's multiplied by 1/3 cubes. You see what we did? They were aware that one times three cubed Sorry, it's one time street you because there was three to the one year. So this is the same as too negative. Two times three. That's two to the power of negative 6/1, all multiplied by one over Twitter world. What is three cubed? 27. Okay, so that's one over to to the part. Negative six is the same as one. Over. That's two to the power of six. All multiply by 1/27. So two to the power of six is gonna be a big number. We know that two to the power of 4 16 so two to the power of five will be 32. So to to the path six will be 64 that is going to equal. It's 1/64 multiplied by 1/27. How could we multiply too big numbers like that. Together we can go 64 multiplied by 27 and we're using our columns. What is seven times forced? 28 we add up to what is seven times six is 42 plus two. That's 44. We add a zero. What is two times for? That's eight and what is two times six. That's 12. So we add these together eight 12 and one, six and seven and one, so that is equal to one over. Sorry. Got a bit messy. Day 1000 728. Don't ever underestimate yourself. You could always figure out the odds that remember, if you think that that's too big, go back to something else we learned which was multiplication by columns. Okay, that's how examples and operations. And I'm gonna just leave this up here for you to pause the video and write down if you want to remember your laws that we went through in the previous videos and some scientific notation. Well done, guys. That's a sexually exponents done. 21. Integers (1 of 5) Ordering and Comparing: Hello, Grid. It's now we're gonna be doing ordering and comparing. This isn't our intelligence section and it's our first section on vintages. After this, we're gonna be doing a video adding and subtracting images, multiplying, dividing and in multiple operations, you'll find your get into jizz pretty easily. This is a subsection of what we're doing with ordering and comparing. And by the end of this video, we would have fold in something over here, right? Let's get started Firstly, before we get into into just what is a whole number? Ah, whole number, as you might have learned in our previous videos, is Ah, hold number. It's a number that you can hold something you could hold three grapes. You could hold zero grapes. You can't hold negative one grape. So whole numbers. Any positive number? Plus, including the number zero, it could be 1/2 or fraction or decimal. Then we have vintages, and in Tages exactly the same as whole numbers. But they include all negatives right there. We have positive integers which isn't excluding zero. We have negative integers which is excluding zero all the way negative on then we have non negative integers which is including zero. You don't need to worry too much about that. You also know we have fractions, which is numbers over one another. As you can see, 1/2 5 7th And then we have decimals, which is quite easy. That is 0.251 point 618 all the way. So that's just to get you. It reminded what images are. So let's start ordering on comparing the most important thing you're gonna learn in ordering and comparing is what this is. Now this is a greater than sign. Okay, so what? I mean by a greater than sign. It's bigger on the side and it's small on the side. That's an easy way to remember it. So we have five is greater than one, and that's because this is a great on the inside, so we can say that five is greater than one. Remember, we always put ah bigger number on the left hand side, and this is what it looks like. So we could also say zero is greater than negative for now. How do you know which numbers greater now? I like to think of it in money. You can always think I would rather What amount of money would you rather have? You would always rather have fiber. And instead of one red, You're also always rather have zero around instead of negative four round. So the bigger numbers always the mountain man you would rather have. So let's have a look at ordering. We have these number zero negative 30 negative two to negative 12. So how could we put these in order? Well, we look at our number line. We've just kept a negative 10 to 10. So what is the smallest number here? If we kept going negative, negative, negative. We eventually get to negative 30 South First number is negative. 30. We put a little comment. What is the next smallest number? That's negative. 12. So you put it negative. 12. Then we have negative, too. We can cost him out as we use them. Then we have zero, cause it goes negative, too. Zero and then we have to. And this is ordering numbers from smallest two biggest every year. We could have done it for biggest to smallest backwards. Let's have a look at how we would compare numbers. Let's take 00 and negative three, which is bigger, right? We're gonna use our sign. Remember, this is saying bigger than and this is saying smaller than okay, so that is bigger than for B. And this is smaller than with IHS. But the bigger number always goes on the sides where that is the biggest. Some people think of it as a crocodile mouth where the crocodile always eat the bigger number and there's a crocodile. Lose I and his teeth, and that's where he will always eat the bigger number. Since Lords or zero is bigger, then negative 30. Let's try that again. Negative too, is smaller than two, so negative two is less than to okay, Negative 12 and negative 20 which is the smaller number? Would you rather have negative 12 ran or negative? 20 rend? I would rather have negative 12 rant than negative Twin red. So negative 12 is greater than or bigger than negative. 20. Let's have a look at this Easy one, which is bigger three or 23 is bigger, so we put it like this. Three is greater than two. What about saving a negative 12 7 is greater than negative 12. What about negative 14 and negative 15 negative 14 is greater than negative. 15. Remember, we could also do it the other way, and I would simply be less than so. Yeah, is let's try some truffles is negative. 12. Greater than 11? No, this is clearly false. It should be that negative 12 is less than 11. So we're gonna circle that is false. Not true is negative. Seven. Less than negative nine. Let's have a look at a number line. We know that there would be zero over here. They will come to negative seven. They will come to negative. Nine Is negative. Seven. Less than negative known, which is bigger number year. The bigger number is seven because seven is the bigger number, it has to have the biggest side off the sideways V. So it is negative. Seven is bigger. So originally, this is also false. Not true. Let's try his zero less, then negative one. So we're gonna have negative one year, which is the bigger number out of zero and negative one zero is the bigger number says zero has to have the crocodile Smart zero is greater than negative one that's not create. So this is again false. Okay, so the things I want you to remember in in, um ordering and comparing is that bigger side. The bigger number always goes at the biggest side and the smaller numbers all on the smaller side. If you see it facing this way with the crocodiles eating twos live like that, we say it's greater than that's how we read it. That's simply how we read it. It's the exact same symbol. So we say 10 is greater than 5 10 is greater than five. And if we read it that way, we ride five is less than 10. And so it's this way. It's less than that way is greater than it's simply a way to read it. It's the same symbol, but the bigger number must always go on the biggest site. It's that easy to remember. Okay, let's move on to now. Addition and subtraction 22. Integers (2 of 5) Addition and Subtraction: Okay, Great AIDS. We still in the Inter just section and now we're going to do addition and subtraction. Uh, vintages we had previously done ordering and comparing. Okay, remain mint. Ordering him comparing we simply learned about this symbol with a bigger number. Always has to go in the biggest side. So 10 is greater than five. If it's facing that way, it's great to then it's facing that way. It's less than how we're gonna do addition and subtraction. We have a really done whole numbers addition and subtraction. You'll remember this when it was 242 past 39. How do we do that we went to plus nine is one. Carry 10 1 four plus one is five plus. Another three is eight. There was nothing there, so it's simply too. That was how we did addition. And how do you do Subtraction? Subtraction is just Azizi. We had 233 minus 277. What is three minus seven? We can't do it, so we need to change that into two and borrow the one to make that 13 13 months, saving a six two months to zero and two months to zero so that Allen's is simply six. But now what? Are we gonna do it if it was negative? 27 at the top. Negative 227 at the top. So what is an integer and integers a negative number? So what if we had 227 minus 223? This answer is gonna actually give us negative six. And we gonna work through a have to do that? No. So we had an example. Like what if it was 242 plus negative 59? That would probably be just the same as that. What if it was 240 to minus minus 39? This is a really interesting, easy one to do, and we can show you how to do that. Now, what about negative 240 to minus 39? Again, we're gonna show you how to do that. So we have a number line. It's always here just for reference for you. So how are we going to do 240 to minus 39 240 to minus minus 39? Sorry, that was 242 plus minus 3940 to minus than unite minus 39 minus turned 40 to minus 39. There's a 240 to minus 39. When we have a plus multiplied by a negative, we're going to find that this is very similar to the multiplication video. When we have a plus multiplied by negative, it's simply stays a negative so we can ignore the plus ve and make that a miner's. So that's 240 to minus 59. Make sure you watch the video multiplication if you don't understand what we're doing here . It is the next video and very worthwhile. So what's 240 to minus 39? We can go to know 42 minus 39 that you can easily do. We can't go three miles. Nine. We borrow three. We go. One. What's 13 minus nine? That's for three months. 30 and that's two. That's very easy. So how would we go? Turner 40 to minus minus 39 when we have a minus in a milers. It gives us a plus. Now, all want you to remember. No, is that it is a minus multiplied by miners. It always gives you a plus. Must watch our multiplication video to figure this out and understand how that works. Okay, so that is very easy to under 42 plus 39. I'm not even going to do that for you. So now we have negative 3240 to minus 29. So that somebody is negative. 240 to minus 39. Now, what you could do here is you can do something where you take the minus out and that becomes 242. Dress 39. Okay, so 232 plus 242 plus 29 all times where miners? Because every times out again, it would simply equal that. So these equal both equal the same thing. So what's 242 plus 39? We worked it out there. We should have, but I'm gonna quickly do it here. Two plus nine is 11. That's one. That's whatever they won. First force 5678 Ace not carrying anything Plaza to, but we still have a miner's over yellow. So we got to bring down on negative every year. So that is equal to negative. 281. It makes sense. If we have a number all the way down here minus 242 and we minus ing another number, it is going to give us a negative answer. So you must have a negative answer. Okay. What about example one, two plus four minus five plus minus five minus, minus three. How are we going to do that? Let's have a look at the number line we started to right. We plus 4 +1234 Right. So we have six now. We minus five. This is still quite easy. +12345 So we on one now, we've gotta add negative five. What do we say when we add a negative five? We can simply say minus five, so we're gonna minus five again. 12345 currently on minus four. And now we're gonna minus minus three. What happens when we minus and miners? We know that's gonna quart plus three. So we go +123 and is our answer. Negative one. So remember, plus goes up the number line and minus goes down the number line. If you were placing and negative number. You would also go down the minus than the Tottenham alone. Let's have a look. An example to negative 10 plus 14 minus five minus minus six minus minus one. This should be quite easy. Let's start of negative 10 although every year plus 14. Sorry, negative. 10. Although every year Class 14 goes 10 to 0 as another Force four minus 54 minus five said this year's four full minus five is minus one. So we have minus one up to you. Minus one minus minus six a minus a minus six. So we want a minus minus six. How are we going to do that? We gotta plus six. So minus one has 123456 week Arnold five minus minus one. What is a minus? Minus one? It's plus one. So we're from positive. Five plus 1 to 6. And that's arm. So now I have negative 20 to minus full, minus minus five. A class minus five minus 53. So they started beginning. We're gonna go negative 20 to minus full. That is equal to negative. 26. We now have negative 26 minus minus five Cem. Minus and negative. Five. Remember your brackets. And that is the same as negative 26 plus five rights. Don't forget that. A minus and a minus is a plus. Was negative. 26 plus five. So we all the way down day has five going up that is equal to negative. 21 can. Where we were Not yet. What is negative? 21 plus and negative sides. That is the same as negative 21 minus five. Which is the same as negative to me. Almost five negative. 26. Right now we have you here now. What is negative? 26 minus minus 53. This is equal to the same is negative. 26 minus and a minus is a plus 53. And that is equal to what is negative. 26 plus 53. So we all the way here. Negative 26. So we all know zero and we're all the way and negative. 26. We're gonna plus 53. We go all this way. Yeah, and that is gonna equal 27. Positive. 27. Positive. I remember. A positive X is equal to a positive eggs at plus minus X is equal to n minus six. A minus minus and X is equal to a positive aches and a negative X is simply and negative AIDS rights. Let's have a look at one more or one set of examples. Negative. 127 miners. 372. How are we going to do this? Right. So the first thing you can see is that the bigger numbers at the bottom ignoring the minus the large and numbers at the bottom. Forget about the miners. So we're gonna swap it around 373 goes to the top 127 comes to bottom. We still gonna put a minus on. So we're gonna drop the miners, though this time because we know that this whole number because the bigger numbers miners, it's gonna equal miners. So let's go three months. Seven. We club deuce, we borrow when we make that six, we bring a one year 13 minus seven is six, six months to is four and three minus one is to 246 negative. 246. Very important to have that negative year. Let's have a look at that again. We're gonna have to 327. Minus 3619. How are we going to do that? Well, we know the bigger numbers at the bottom. We can see that once is going to be negative. But we've got to switch it around, so we're gonna draw thing here. We're gonna go 303,619. Take away 327. What's the most important thing? We've gotta drop on. Negative dome. What is your A minus seven record to that? Let's borrow one. Make that aids. Make that A 10 10 minus seven is three eight months to 66 minus three is 33 months. Zero is three. And that is our answer with the negative. Okay, let's have a look at those. So how would we do? Negative 127 minus 477. So we simply gonna put on negative 127 at the top and our 477 of the bottom. But we're gonna add on negative. I remember this to the same is 127 1 a plus 1 477 Because remember, these two numbers can be the same. Was negative. 127 plus 477. Don't forget that, but we still have a negative. So we've got a drop on negative. Seven plus seven is 14 and there's a one seven plus twos. Nine. Passive one is 10 we add a zero has a one. One plus one is two plus four is six, 600 full Great. So that was addition and subtraction. We learned that classics is equal to plastics. A plus times a minus X is equal to minus six on minus times a minus six Secret of plastics and a minus six equals Ammon Itics. This is very similar to multiplication, but we're really gonna go through this in much more detail in our multiplication video, which is next 23. Integers (3 of 5) Multiplication: Okay, everyone. We now in a video, multiplying and dividing off into jizz, right? We're gonna have to separate videos, one of multiplication one and divide. This one is currently on multiplication. If you watch the audition video, you have seen that we use very similar rules. In addition, we couldn't get away without showing you how this works in our multiplication. So we're gonna go to that right now By the end of this, gonna fold in some useful information here, so multiplying in columns. You've already done this with whole number multiplication. You went 273 multiplied by seven. That was easy. You said seven times three is 21. You knew your timetables. Well, you added up to He added, Your 27 times seven is 49 plus two is 51. So there's a 57 times two is 14 plus five is 19. I never put on 19 in here. Now we added are meant we didn't you need at a magic zero. And that is the answer. And that is the answer. 1911. You also knew how to do those two numbers. You said Turner 42 multiplied by 33. So you went three times two is six. Three times four is 12. It's good to practice this. We carry out 13 times. Two is six plus one is seven. We add a magic zero three times two is six, three times four is 12. So two time at another one. Another one three times two plus one is six plus one is seven so six plus zero with draw a line. We now make it a plus. Six plus zero is six to plus six is age seven plus two is nine and in seven and there's no answer. But now what about images where we have negative numbers like this? What if we said 242 multiplied by negative 33? How are we going to do that? Let's look at some rules and these are similar to the rules we used in addition and subtraction. So one times one is equal to one that's easy. Naked of one multiplied by one is equal to negative one one multiply by. Negative one is also equal to the same as negative one times, one which we know is negative one What about negative one multiplied by negative one. When we have a negative number and a negative number, it always equals positive, and that's positive. One. So this could be said as X, or what this can be said is why multiplied by eggs and that is equal to y eggs Negative y, multiplied by eggs, is equal to negative Y X. The same, of course, would be true for why multiplied by negative eggs to equal negative Y X, it's the exact same answer and negative y multiplied by negative ICS is equal to Y X and multiply or negative times. A negative is equal. Don't positive. So negative number multiplied by a negative number is equal to positive, and a positive number multiplied by negative number is equal to negative can. Let's practice some examples. We have 73 multiplied by negative seven. What sign is the? It's gonna be what's a positive number multiplied by negative number, positive times by negative equals negative. So we know that this is gonna have a negative Let's do this quickly. Seven. Multiplied by three is 21 so that's what one plus two saving multiplied by saving is 49 plus two is 51. Are we done know it's a negative 51 because a positive multiplied by negative is a negative . What about negative 42 multiplied by negative 33? Let's quickly figure out what is a negative number multiplied by a negative number. Remember, we could say negative one multiplied by negative one, and that is equal to positive one. So the answer is going to be positive, said 202 times, three times to six. Three times four is 12 so we add up. 12. We had our magic zero three times to his six. Three times four is 12. We add these together six past eras. Six to plus six is +81 plus twos, three and then one. And what do we do here? We put a negative. No, it's a positive, but we can just leave it as it is, and that is our answer. So let's try and negative 14 times. Three what's in negative, multiplied by positive that is equal to and negative. Sometimes people write this *** to have times a positive, and that is equal to a negative. That's a V and the just for interest. So three times four is 12. We had a 13 10 for minus three plus one is 4 42 that I want to know. We've gotta add and negative. And that is our answer. Negative. 42. Okay, so we learned Multiplication and negative times and negative is equal to a positive and a positive times and negative is equal to negative. A nice way to remember this is there's always two negatives and one positive in the equation. No matter what way you put it, it could even be negative times. A positive is equal to negative, but there's always one negative two negatives and one positive. Here's our one positive and one negative and two negatives. Okay. And that is multiplication. Now we're gonna go into division, and we're gonna look how that is exactly the same as multiplication. In some ways, 24. Integers (4 of 5) Division: can always still in inter Jizz and we now doing division. We're gonna fill in something at the end here, and we're gonna see why Multiplication and division are so similar. Okay, let's get started. So you've already done whole number division where you said seven divided by 223. So nice way to think about this. This is the house we have here, and saving is trying to get into the house. Can seven get into to No. Seven can knock it into two. So that 00 times seven is zero. So we bring down about two or two minus zero is too. We bring down our other two and no f 22 can save and get into 22. Yes, it can. How many times? What is seven times 3 21 So we know I can get in three times three multiply Its by seven is 21 21. Now we two minus two is zero to minus. One is one. So I should've done it that way, and they would bring down a six like we brought down to and we have 16 can save a going to 16 years is everything going to 16 twice, two times seven is 14. So we had 16 minus 14 and that we know is equal to two. So that is 52 Remainder two. And then you try 21 into 226 can to anyone going to to knows. So we would pray zero there. But let's just go. Can 21 going to 22? Yes, it can go once. So we have our magic. Zero. We'll know about it. One times 21 is 21. So add that day we just skipped a line. That's fine. Tu minus one is 12 months to zero six sprinted down is six. So we have 16 can 21 Gonna 16? No. So this goes in zero times, then remainder 16. Which is that over there? So that would be 10 remainder 16. Right? So you remember this. But now what about images? What about negative numbers? What if we said How many times is 21 going too negative? 226. It's the vision, not division is simply a special case of multiplication. So if I said 20 divided by five or 20/5 that is the same as saying 20/1. Multiply by 1/5. It's exactly the same. So we could simplify this like you've done infractions. And as 20 multiplied by 0.2 because of 1/5 is equal to 0.2. And that is you got to fall. And as you could tell here that 20/5 was full. And that is why multiplication division is so similar to multiplication. So what about 217 divided by seven, We can say that's 217 divided by one multiplied by one of the seven, and that you can subsequently workout is 31. But now what about images, including negative numbers. So we've just shown that division is the same, but as multiplication. But so what are we doing with negative numbers like negative 217 divided by negative seven . Well, if it is exactly the same as multiplication, surely we're gonna ply the exact same rules because remember X divide about why is the same as X over one multiplied by one ever Why? Okay, When we multiplied indigenes, we found some rules. One times one is one negative one times one is negative, 11 times negative one is the same as negative one times one, and that is equal to negative one and negative one times negative one was equal to one. Remember, these were multiplication rules and negative number times. A negative number is equal to a positive number and a positive number. Times a negative number is equal to negative number. So the vision is simply a special case of multiplication. So let's go. What is negative? 20 Divided by five. That is the same as 20/1, multiplied by one of a five. And that is negative. 20 over negative to and what's a negative multiplied by negative and that is equal to negative. Full? What about negative? 217 divided by a negative seven. So we go to negative turn in 17 divided by negative seven and we got simply are 31. What is a negative times? A negative over here? Negative multiplied by negative, and that is equal to a positive 31. So what about a negative seven divided by 226? What do we know that I was gonna be It's gonna be a negative. We could tell that before we've even started so many times a seven going to 22 3 times three times seven is 2121. How many times we ignoring the negative knocks, you know that I was gonna be negative three times seven is 2121. That goes 01 16. How many times? A seven under 16 twice, two times seven is 14. We vote out of negative six and then 16 months. 14 we know is equal to two. So that's remainder two. So it's negative. 32 remained, too. What about 21? Divided by negative Tune into E six. Again, we know the answer is going to be negative. We can ignore everything else. So let's just ignore that now because you know that I was gonna be negative. Leave it there, though, that ignore it. 21 goes into 22 once. One times 21 is 21 20 to minus 21 is one we bring down on 16. We say 21 times goes into 16 0 so that's a zero. Let's extend that 10 remainder 16 but importantly, it's negative. 10 remainder 16 over. Yes, a negative. 10 remained a 16. What about negative 21? Divided by negative 22. So this is the same as negative 226 divided by negative 21. And those are gonna cancel each other out nicely to make it both positive. So 21 into 22. We know the answer is 10 remained a 16. It's exactly the same as this, but it's a positive now. Okay, Well done. The things you need to remember is that it can be treated exactly the same as modification . The same rules apply because X over. Why is the same as X over one multiplied by one over wire. Now we're gonna go into some examples of multiple operations. 25. Integers (5 of 5) Multiple Operations: Okay, Great. It's now we're gonna be doing still images the last section of video in the section on multiple operations. By the end of this video, we're gonna fold in some information here. Some multiple operations. Let's go straight into some examples. What is one multiplied by one multiplied by one that is simply equal to one. It's the one operation twice. What is one multiplied by negative. One multiplied by one. What is one multiplied by negative one. We know that that is equal to negative one. So then what is negative? One multiplied by one. And that is equal to negative one. Negative one. What is native? One Most played by negative one Most lima negative on most private. One negative times a negative is a positive positive times. A positive is a positive. That is equal one. What about negative? One look. Clever. Negative. One multiple of a negative one So negative one times a negative is equal to positive one and positive. One multiplied by negative. One is equal to negative one. Make sure you understand how this works. Okay. What about negative One divided by one negative. One of one is the same as negative. 1/1 multiplied by 1/1. So what's a negative? Multiplied by positive and that is equal to negative one. So negative one of a negative one that is the same as negative. 1/1, multiplied by one over negative one. So negative multiplied by negative because the negative can be in the top or the bottom. And that is simply equal to one. What about negative? One of a negative one multiplied by one. So we just calculated that that is equal to one one times one is one. Okay, now we got a real example. We have negative 352 divided by 32 last 51 multiplied by negative. 27. We can see there's gonna be a multiplication, and the divisions have put easier for us. But we've got to remember bod maths. Now, what is barred mats? Remember it from from junior school and previous years, brackets for be order, divide, multiply Adams of draft remember brackets and orders to to the power of four. For example. We start with these two brackets in order, Then we move on to division of Multiply and then Adams attract. Do we have any brackets. Do you have any order? No. Do we have any divide or multiply? Yes, we got a division animals application. So let's start with those two. 51 multiplied by negative 27. Okay. Now, what is a positive multiplied by negative. We know that answer is gonna equal negative. So let's ignore that for now. Seven times one is 77 times five is 35. We had a magic zero two multiplied by one is too to multiply by. Five is 10. So let's add these all together with a big plus 7 +731 Is that answer? No, it has to equal negative. So we're gonna say negative. 1377. So pop, plus negative. 1377. Okay, now we're going to say out division. 352. Negative. 352 divided by 32 way. Does the strain of 50 to go and wait Is the 30 to go here? That number the top always goes into the house. So negative. 352 divided by 32. Now what is it? We know that negative is a special case of multiplication. So what is a negative number divided by a positive number? We know that's gonna equal and negative number so we can ignore the negative here for now. How many times? A steady to going to 35? That's once. One times 30 to study, too. 35 minutes 32 is three. Bring down the two. That's two. How many times? A steady to going to 32 once So this is equal to brackets and negative 11. What is a positive multiplied by negative? We know that he's got equal to a negative. So this is a legal to negative 11 minus 1377. And what is that gonna equal That is simply negative. 1388. Well, then, let's try another beast of an operation. We have negative 31 multiplied by 15 or divided by 12 plus seven minus minus five past times Negative. Seven. We know there's a multiplication and division here, and we must remember botnets are then your brackets. Yes, there's bracket and a multiplication that brackets doesn't even need to be here. These brackets don't need to be or they do need to be. So they start with the multiplication. What is 31 times 15. So I'm gonna say negative 31 multiplied by 15. What's the negative times? A positive. We know that's gonna equal negative number Negative V. So let's say that's five. So we don't need away about that negative for a while, so we know that I was gonna equal negative. So five times one is 55 times three is 15. We add a 01 times. One is one and one times three is three. We add us together to get 56 full. Is that on set? No, it's negative. 465. So we know that that is gonna be equal to negative. 465 at the top, divided by what's 12 plus seven. That's easy. That's 19 that is equal to minors. Negative. Five multiplied by negative seven and negative times. A negative is equal to a positive. Five times seven is 35. That's positive. 35. Okay, so now how can we do this year? Negative. 465 divided by 19. That's easy. We can put that here. Remember the negative 465 and the 19. Remember, the bigger number on top always goes into the house. What's a negative? Divided by a negative. That is what a negative divide about positive that's gonna equal a negative. So we don't need to worry about that now. We know I was gonna equal negative. How many times is 19 going to 46? 19 times two is 38. What is 19? So let's write that down 19 Multiply by two is equal to 38 and 19 multiplied by three is equal to 57 0 we can see is not gonna be that. So it's multiplied by two. So two times 19 is equal to 38. What is 46 minus 20 minus 38 then we ignoring that that is equal to 80 And again, let's go six type minus eight We, but we take a three. We borrow one to take that to three. We got 16 months eight and that's eight. And in three months, 30 we pulled on a five and we get 85. How many times is 85 go into or 19 going to 85. Okay, what is 19 multiplied by five 19 multiplied by five is five times 19 is 50 foot plants five times nine is 45. That is equal to 95. So doesn't go in five times in Moscow and four times what is 19 multiplied by four that is equal to 40 plus 36. That's 76. So we put a 76 every year we minus the to six month at five months. Six record. Do we borrow one to make that seven and make a 15 minus 60 Quilty nine and that is a zero. So it's 19 remained a nine. It's 19. How many times there's 19 going to 90 times, so 19 cannot go into nine, so we got 24 remainder nine 24 a man and nine weaken. Subsequently say that is negative. 24,000,009 minus 24 remained in nine minus positive times of a negative, minus times a positive and negative times. A positive is negative. 35. So what is 24 negative? 24 minus 35. Two Negatives is staying negative in this case because is minus plus, um, miners, so it could pull the minus ounce. So negative 24 May 9 minus 35 is minus 69. Remained a nine and that there is your answer but complicated. Don't stress too much about it or the nitty gritty. But you understand how these multiple operations work. Make sure you understand the fundamental year. So for multiple operations, we must've will get up bod mats where we have brackets, order, divide, multiply, add and subject. We have our bod, miss. It's important to remember and body Mass that it's being our first Denham next and a and it's lost. Well, then, that's our section, about minutes. 26. Common Fractions (1 of 8) Simplifying fractions, including mixed numbers: Okay, Great AIDS. Now we're gonna be doing fractions and we'll start off in this video by simplifying fractions. As you can see afterwards, you're gonna add and subtract, multiply and divide mixture. Improper square and cube roots percentages adjust just pie percentages and word problems. So let's get started with simplifying fractions. So what is a fraction is the first question. Imagine you had a 50% sale at the shop. This is the same as 1/2 cell, but we never say half that doesn't really work. We just say a 50 Pinsent sale, but 50% is the same as 1/2 right? So a desk more percentage in a fraction off the same way to represent a number. How do I mean that 0.75 is the same 75%. 75 over 100. That's how we simplify it. 75 of 100. But now, with fractions, we're gonna get into how we convert these or around in the future. But infractions, you've got a simplified. That's not the simplest form of a fraction. For example, 75 of 100 Could we divide the top and the bottom by two 75 divided by two is not going to give us a whole number. So we try. Divide by three again. It's not going to give us a whole number is divide by four node won't give us a whole number. If you put them in a calculator, you'll see a lot of decimals. What about divided by 5 75 divided by five? That is gonna work. So if we divide both 75 100 by five, we get 15/20 and that is to whole numbers. But still, we can simplify this even further. So let's go. Can we divide by two carry divided by three? No. Can we divide them before? Nor can we divided by five. Yes, we can divide that by five, and we're going to get three over full. And that is the simplest form of that fraction. Three other fall is exactly the same as 75 over 100 he said. Three of a four is a simplified fraction. So do you remember how to do highest common factors? And that's the hot highest of the house. Highest coming factor. H I G h years t highest common factor right. So what is a fraction? Okay, let's start off with what is a fraction A 75 of 100 right? Who did that? So what are the factors off? 75 100. So what are the factors of 75 Remember? Do you remember factors? If you can't remember how to find a factor, make sure you watch the video and factors. The fact is, 75 or 135 15 25 and 75. All of those condemn vied into 75 to give your whole number. What about 100? We have a lot more. We have 1245 10 2025 50 and 100. So now the first step is done. We found our factors. What are the common factors off 75 100. So what are These? Are the same. We can immediately see that 15 and 25. You see that one? We both have ones. They're both have a five and we both of 25 25. No other, no other are common. So what is the highest common factor of 75 or 100? And that is equal to 25. So what do we use of this? So divide both numbers by 25 your highest common factor to simply find the simplified fraction. So 75 divided by 100 divided by its highest or highest common factor, divide by the top of 25. We divide by the bottom by 25 that gives us 3/4. So the high school, in fact image that will always work. But it is often easier to take a couple of steps to get those. So what I mean by that? We'll get into that in the examples. Let's have a look at a mixed number. So firstly, we had 75 is equal to 3/4. That was easy. What about 100 over 75 that is equal to 4/3. So for over three, and that is the same as 3/3, plus 1/3. So that is, that would be the same as Let's just write this in extended. That would be the same as 33 plus one rise, and that is the same as one and one of the three. So now we have a mixed number. Why is it a mixed number? Because it made up of a whole number and a fraction. How do we get back to the mixed number? We go three times one. So it's we first, we know it's gonna be over three. That's the first thing we do now. We want to find the top number. It's three times one is three plus one is full, and that's how we go back. So that times that is three past one is full, and that goes at the top. What about 95/45? Well, 95 of 45 is the same as 19/9. How did I get that? What if it divide each of the top and the bottom by five? We divide by five. We know that 45 out of 59 and 95 divided by five. That's how is difficult, but we know that 100 divided by five is 20 so 95 is just five laces. 19 that is. The same is nine plus nine plus one, so that is equal to 19 all divided by nine. And that is the same as two and one knife. See there were two nines of yard is one and two, so we took those to make it two times and one millions. So how would we get back, we say to multiplied by known Well, first we could. I'm 900. Then we say, to multiply by nine plus one, so to multiply by nine is 18 plus one is equal to 19 and that's how we got on 19. Let's have a look at examples is the best way to practice and learn. Let's start by simplifying these fractions. 64 16 divided by 64 Now remain. We could go and find the highest common factor, and you could probably find this quite easy. But what else you can do is you just can't keep dividing. Can we divide these by two? Yes, we can What a 16 divided by two that is equal to AIDS. What is 64? Divided by two that is equal to 32. How did I know you could divided by two Because they both even Can we divide that by two again? Yes, you can. You in luck. Eight. Divided by two is four 32 divided about 2 16 Can you divide it by two again you can. Four. Divided by two is to 16 of our but two's eight, and you can divide about two again to get one and eight. Divide about two years for and that is your answer. What about 27/30? What is a common number here? We know that 30 could go into 3 10 times and to 27 is three less and 30 so that must go in nine times. So that is the same as 9/10. What is negative? 56 Divided by negative. Oh, divided by 76. We've got to keep the negative. The whole time is thought by having it. What is 56? Divided? About two we know 50 divided by two is 25 60 Viber twos. Threes 25 plus three is 28 so we have negative 20 AIDS. What is 76? Divided by two We know 70 divided by two is 35. Then the 62 vital matters 3 35 plus three is 38 and we can divide those by two again. Remember the negative and there's negative. 28. Divided by two is 14 38 divided by two. We know the city divided by two is 15 8 Divide, about two is 4 15 plus four is 19 and that looks about as far as we can go. And that is aunts and negative. 14/19. What about 17/34? You can quite easily see here that 17 times two is 30 force that goes in 1/2. So that's how we simplifying fractions. There's not Try some mixed numbers. So 112 divide about 64. Okay, so we're gonna put us 64 the top here, and we can see that 64 goes into that at least ones. So we're going to say 64. So what's 112? Minus 64? Let's do that Over here. I'm gonna do right at the top 112 minus 64. Or let's go. Rather minus 64. How many times does to go minus for your client? You're going to take that to zero. Borrow one, and there's 12 12 minus four. AIDS 10 because we take the one minus six is full. So we know that, plus 48. So that's plus 48. So we know that that is equal to one and 64 over for 48/64. We can simplify this, so we keep it as one. Why do we have both? Those? The top of the bottom 48 divided by two, is 24 64. Divide by 2 32 We can have it again. Ever get one and 12/16? We can have it again. Um, and write it here. 1 12 Divide about 26 16 divided. But Tuesday's we can have it again. And that is one and six, divided by 238 divided tattoos for so one and 3/4 are answer. What about 30 Divided by 24 So we know that is 24 then 30 is the same is 24 plus six, which is the same as one and 24 Bovis with six of the top because we can see 24 plus six one times 24. We can divide both the top in the water about two, so there's one and 3/12. Can we do anything for the are we can divide, but the top of the bottom by three no. One and 1/4. And that is our answer. What about 76,000 divided by 13,000 short. One easy way to do this is you can simply cancel out zeros. If for every zero on the top you can cancel out is here at the bottom, starting from the rides. So that is equivalent of 76. Divided by 13. Can we simplify that? Any food and our 13 years of prime numbers. So we need to see if 76 can go into 13. So let's try. What is I'm gonna write severe 13 times five. That is equal to 50 plus 15. That is 65. Okay, what about 13 times six. So we simply gonna have to add another 39 to that. So it's 16 past 13 is 78th s, so we can see it's not gonna divide there. So that is simply our answer. And that is OK. What about negative 81? Divided by 63. It's going to stay a negative. Anyone divided by 63. What is simple here? What do we notice here? These are both divisible by nine. How do I know that because the beauty about the nine times table is the numbers add up to 98 plus one is nine and six. First three is nine. So the same 81 divided by nine is nine. Because you know that nine times nine is anyone and what 63 divided by nine. And that's seven. And now we can go further and say Right that is equal to seven negative seven plus two because nine is the same seven plus two. So we can say that is negative one and seven and to over seven. And that is our answer. Well done, guys. That is how to go from mixed numbers and back. Just remember, if you wanted to get the other way, said there was a number three and four or three and 3/4 you would say to get from the mixed number two. The fraction you say four times three. So four multiplied by three after. Of course, putting for the bottom four times three is 12 and 12 plus, the three on the top is equal to 15. And that would be your answer. 15/4. But what if there was four and 1/3 would say three times four is 12 plus one is 13 and that's 13/3. And that's how you go the other way. That's how we simplify common fractions. We're gonna do a lot more mixture and proper soon, so don't stress too much about those last section. 27. Common Fractions (2 of 8) Convert Mixed fractions to improper: Okay, Great dates. Now we doing mixed to improper fractions. Now, you might notice that this is actually the second video in the series, and this goes under simplifying fractions. I wanted to move this to the second video of Syria's because I think is extremely important to understand this before you go into addition, subtraction, multiplication and dividing. And if you watch the simplifying video we lead and straight from the simpler vying for your into mixed to improper fractions, right, let's get started. What is a proper fraction? An improper fraction in them extraction. A proper fraction is where we have, for example, 3/5. And the top number is smaller than the bottom number, which is bigger, so proper. The top number is smaller. The bottom is bigger, easy to remember. Improper is where the top come is bigger on the bottom number is smaller again. That's easy to remember. Mixed fraction is where we're gonna have a fraction or a whole number and a fraction. So we want to commit very mixed numbers into improper fractions and vice versa. For example, 17/5 that is the same as five plus five has five plus two, that is, you know, 17 is 55 is 10 past 5 15 plus 2 17 all divided by five. That is the same as three multiplied by five plus two. Cause there's three fives, all divided by five. So that's three and two foots. So how would we go back? We have three and 2/5, and that is the same as three tons five plus two of five that is 17/5, and that's the same as three times five. Keeping the nominated the same, you could see how that makes it so much easier if you the bottom number is always the same . And we have three times five, which is 15 plus two, which is 17. Because that over here is equal to 15. It's going to example because that's the best way the load we're gonna convert this mixed number remain. It's a mixed up because there's a whole number of fraction into an improper fraction. So what's four times to that is eight plus one over there is nine, and that's 9/4. Okay, one and 2/3. We start off by putting our 33 times. One is three plus two is five. Okay, negative. 7.5. We started by keeping up to We've got to keep our negative. Two times seven is 14 plus one is negative. 15/2, one and 2/33. We keep our 33 and we go 33 times. One is 33 plus two is 35. Okay, great. If you really want to learn this well, tried the question. Pause the video and try the question yourself, and then I'm gonna do it for you. There's another formal questions. Five times four is 20. So we keep the five. The bottom five, since Force 20 plus one is 21 que negative seven and three of a full. We keep the force the same four times. Negative seven is 20 A's plus 3 28 Mastery is 31. But wait, we got to keep our negative 10 12 and nine tens. We've got to keep our 10 10 times 12 is 120 plus nine is 129 six and two of the 33. An easy one tree and off with three of the bottom three times six is 18 plus two is 20. So that's 20/3. And that how is how we did mixed to improper fractions. Now, remember, we added that over here because we thought it would be a much easier for you to understand year before we're gonna go into addition, Subtraction, multiplying and divide. Well done. Great. That was really impressive stuff. 28. Common Fractions (3 of 8) Adding and Subtracting including mixed numbers: Now we're gonna be doing addition and subtraction off common fractions. Remember, if you're watching this video and you haven't watched any other videos in the section, we did the mix to improper video before addition and subtraction. It is so important to watch this as you will see that there's some parts that apply to converting a mixed number two an improper number before you can add and subtract. So if you don't know how to do that, make sure you watch that video first. Okay, let's get started. We have half a pie or half a circle. We want to add another half a circle. What a to half of pies. We went 1/2 plus 1/2. What would that equal? That would equal one or 2/2, and you can see that that equals one. So two of the two is the same as one of one, which is the same as one. So what if we took an entire pie and from that entire circle pie we took away half of it. So we had one of the one, and we took away 1/2. It's convert that 1/1 to 2 ever to to make it easier to minors. The half and that is equal to two minus 1/2. As you can see, it's a common denominator, so we can put it both under the same common denominator thing. Go to minus one, and that is equal to 1/2 which you would probably have guessed. So we're gonna have five steps to solving, adding and subtracting fractions. The first step out of five is to find the lowest common denominator the L C D, as you may already have done and learned. So let's take the example 1/2 plus 3/5. So the lowest common denominator. First, we've got to find the denominators of to as 2468 10 12. You could keep continent forever and then five. And that's 5 10 on the Stop it, Tim, because we can see that there's a common denominator and it is the lowest one. So we found a lowest common denominator number to change the fraction denominator. So we're going to go one of the two mastery of a five. Now we want to change that to have tens, so to get us to to attain we're gonna have to multiply the bottom by five because we multiplied it. But And by five, we're gonna multiply the top by five. To get the bottom one to data after multiplied by 25 by two equals 10. So we're gonna have to times a top back to three times two equals 10. So 5/10 plus six of it'd number three and easy step merge the fractions over the common denominator. All you gotta do is merge the two because there's a common denominator. You can simply write them both over. The same denominator then before is also easy. Solve a fraction 5/6 or five plus 6/10 is equal to your level over 10 and number three is simplify. And in this case, 11/10 you could simplify it as one and 1 10 but you might not have to do that. It could be an answer. And as you can see, 11/10 can't be simplified anymore. And you really watch the first video in the section on simplifying fractions to, you know, to do that. Okay, let's look at some examples. We're gonna go through the state one of finding the lows could not common denominator the LCD step to change effective denominator. Step three. Merge over the common denominator. Step four sold state five. A simplified. So let's take this example to Over five Ministry over one. What is the common denominator? You can quite easily will be able to find that the common denominators. 15. It doesn't have to be the lowest common denominator, but it must be a common denominator. If you took 30 the answer would still work. So how do we get 15 returns? Five by three every times three by five. So three times the top day by five on the top. There about three. That's easy. So we changed. Effective denominator. 15 15 3 times two is six and one times five Is five merger of a common denominator. Solve the fraction and simplify, and we can simplify that any further. So 1/15. Let's try another example. We got one on 12 ever four minus one and 2/3. Okay, now, I never I should have added steps. Zero convert to improper because we're gonna have to change his tune. Improper fraction. So we're gonna go 12 before we leave it as is minus. How do you get that? Three times. One plus two. So is over three. Three times one is three plus two is fire. Okay, you could probably see that. You could simplify that, But for this example, we're not going to do that because 12/4 is three. But let's go. What is a common denominator? You can quite easily see that. 12 you could four times three is 12. So we're gonna write 12 and 12. Okay. How much? We got a time to told my three and five by four. So 12 times string is 36 and five times full is 20. And these a minus does not forget. And that is now equal to over the same common denominator. We merge the fractions 56 minus 20. And that is the same. We solved the question 16/12. Can we simplify this? Yes, we can. Is dividing both by two. We get AIDS over six. Well, we could divide them again by two. And that will give us full over three. And that is our answer. Let's try this one. A little bit more fractions in this time So we got 12 04 miners to over three plus two and said 7/2. That 12 before is the same as 4/1. So we're gonna do that straight away for over one, and that is gonna be minus two Service three. And now let's convert this improper Teoh a converting improper fraction so we know it's gonna be plus something over to two times two is four plus seven is 11. What is that? Lowest common denominator weaken. See, that's gonna be six. So let's say that that equals six. How do we get O minus something over six plus something over six. So what time's the bottom? By six to get to six times a top by six. So four times six is 24 three We times about 23 times a top by two. That's for to We gotta ties by three to get a 6 11 times three is equal to 33. So let's now solve a fraction. We added all over six 24 minus forced. 20 plus 33 is 53. Can we simplify that any further? Could be divide that by three or two or sex it doesn't look like it. So that is our answer. Let's go through some examples. What is to service seven has two in one food. So we're gonna convert the improper to over seven plus three and three times to his six plus one is seven now. It's so important to get good at this by understanding what your multiplication czar, another very easy way to find a common denominator, is simply take that multiplied by that. So I always work where you can do that. So let's do that. Seven multiplied by three is 21 21. So we know that that is a common denominator. How many times does seven going to three? I'm sorry to get from 7 to 21 multiply by three, so we're gonna have to multiply by three. We multiplied by seven, so we're going to multiply by seven. So that's a multiplied by three. So to multiply by 367 multiplied by seven is 49. Now let's add them of a common denominator. I'm the skipper Step year, which is also fine. That's gonna be 55. Can we divide 55 by to know by three. No, by seven? No. So it doesn't look like it can be further simplified. And that is also let's have a look at one time, several over four plus 22 to minus four. Every five case. Let's convert us to improper. It's gonna be a four. Just ignore that. That's a plus. If anything, four times one is for her. 7 11 to over two is the same as one of the ones that is Do that minus four or five. What is the common denominator we could simply times four times? One is for for times five is 20. So let's make it 20. So we times by five to get to 11. So that's 55 and I'm gonna show you hear different ways. It's 11 20 four times five to get 2011 times five is 55 20 times one at one times 20 to get 20. So one times 20 of the top, and that's plus 20 miners. We have two times the five by four, so we're gonna have to times before by far for four times four is 16. So what is this gonna equal 55 plus 20 or rather 20 minus 16 is four, so 55 plus four is equal to 59 20 and that again doesn't look like you could be simplified any further. So that is our answer. Let's have a look at one last example. 12/3 minus three and two other three minus 17 over to 12 over the three is the same as for 12. Divided by three is full, but we can just write that is 4/1 miners three times three or it's all over three. Three times three is nine plus two is 11 minus 17 over to what are less common denominator that is equal to six, as you can quite easily see. So one times, multiplied by 64 multiplied by 63 multiplied by 2 11 multiply but to through to multiply by 3 17 multiplied by three for most part by 6 24 minus 22 17 multiplied by three Is minus 51 . Now can we work this out? 24 minus 22 is equal to two to minus 51 is equal to minus 49/6 and can 49 be divided by three or two or six. Note it doesn't look like it. So that is our answer. Well done. That's how we add and subtract fractions. Remember, if you were battling, make sure you do the mix to improper. And now we're gonna go into multiplying and dividing well done, great AIDS. 29. Common Fractions (4 of 8) Multiplying and Dividing including mixed numbers: Okay. Great AIDS. Now we're doing multiplying and dividing off common fractions. Love. This is the first video you watched in the section. We did the mix to improper fractions. Second, we decided that it was very important to understand how to do this correctly before we moved on to addition and subtraction, multiplying and dividing. So make sure you check that out. If you haven't already. I can't. Let's start with multiplying. Negative two and four firsts and times negative. Four and 56 So the first thing we gotta do is converted to an improper fraction. How do we do that five times to get first? The It stays negative. So it's gonna be over to multiplied by something over six and they're both They're going to stay negative. I'm sorry. I put it to your five stays over five. Five times two is 10 plus four is 14 six times four is 24 plus five is 29 right? And then we multiply the top and multiply the bottom, as you might see. What is the negative times? A negative. We know that's a positive. So the answer is going to be positive. What is five multiplied by six. That is 30. What is 14 multiplied by 29 To get 14 multiplied by 29 we're gonna have to do long multiplication. Nine times forced 36 at the 39 times. One is three and nine times one is one has three is 12 at our Magic 02 times four is AIDS and two times twos to its attitude. Together six plus zero plus Take the one because that's 10. 1234 406. And that's 406 and what we can do now we can simplify this, so that is equal to positive. 406. But we allowed to say divide both by tours 203 divided by 15 and Kerry Simplify further. I don't know. We con divided by five controverted by 15. Can we divide that by three? 200? Is that 200 divisible by three notes? Not so we don't, sir. Turn in three won't be divisible by feet three. So that is our answer. Let's have a look at another morning. It example. So the negative 7.5 Times Street and 1/3 so we go that's comin over to, and that's a negative. And that's multiplied by positive Over three. Two times seven is 14 plus one is 15. Three times three is nine pass. One is 10. What is the negative times? A positive? You've learned this already? That is a negative. What a multiplying about him to multiplied by. Three is six and 15 multiplied by 10 is 150. We can simplify this further by dividing and both by half, which gives us 75/3. And can we divide 75 by three? Yes, we can. And that is equal to negative. 25. Don't forget your negative. Well done. OK, let's not look at some dividing fractions. So negative, too. On 3/5, divided by negative four and five of six. So the step one is converted to improper. We know it's gonna be a five. We know it's gonna be a negative. We know it's going to be a divides, and we know it's gonna be a negative. And six. What's five times two is 10 plus three is 13. What's six times full? 24 plus five is 29. Right now we have an extra step known multiplying were two steps non dividing. We have three steps and that extras that comes between step one and three. So we gotta flip the fraction so it's gonna be negative. 13/5 multiplies by negative 6/29. So this is what you got to remember. When you do multiple on division, you flip the fraction and that requires two things. You gotta flip one side of the fraction and you can change Division two multiplication over there. Okay, so this is the most important part of this video Now we could do it easily. What's a negative times? A negative. You know, that's a positive. What is? Five multiplied by 29 right. What is five multiplied by 29. We're gonna have to use non modification 29/5. Five times nine is 45 at a full. That's 10 plus 14 145 on the button. And it's do long multiplication again for 13 times. Six. Unless you can work it out by yourself. It's grades 18 plus one is six and 7 78 Can we simplify this further? It's difficult to figure out if we can simplify this further. We can see we conserve. I'd about two curry divided by three. It doesn't look like 1 45 or divide by three. Could be divide by four. Probably not with five there or definitely not. And five No. Six. So you can keep going to figure out if we can divide to keep trying to see if you get a whole number. But otherwise that is still correct. And that's our arms. Apologies. What? I did years. I forgot to write. Step three, we multiplied the top of the bottom. But I did that over here. So let's have a look at another example. We'll go through it step by step, converting improper fraction. Okay, We know is gonna be negative. Three, uh, divided by five. Positive. Three times seven is 21 plus one is 22. Three times five is 15 plus two is 17. 17. So we're gonna now flip the fraction so negative 22 divided by three Flip. The fraction requires two things last to do. The first is change it to a multiplication, and the second is actually flip the fraction. Okay, And now we're gonna multiply the bottom and the top three times 17 is 51. Firstly, what is this is positive or negative? Negative and five times 20 to 110. And again, you can try and simplify it if you want, But you probably was struggled to find a simplification. And that is good enough is your answer. Remember the negative. Let's do some examples. That's the best way to learn. Negative three and 3/5 multiplied by negative 5/6 case, and it's going negative. We first convert to improper five and then that's gonna be multiplied by negative 5/6. So we're converting to improper three times five is 15 plus plastering is 18 okay, negative, multiplied by negative remembered multiplication is just two steps, so that's step one. Now step here is multiplied out. Multiply the top of the bottom five times six is 30. It's going to be a positive because negative, multiplied by negative, is a positive. What's 18 tons? Five. We know that 18. We know that 20 times five is 100 so now we got to less five times. That's 10 lives. How does he quit in 90? What is 90 divided by 30. We can simplify it. And that is three. Because we can cross the zeros in the 95 by three. Let's try another example. What is negative? 1/7, divided by 43 of Inter Mystery. First is converted negative. 1/7, divided by over three Positive three times four is 12 plus twos. 14. Let's flip the fraction this is now stick one. Now Step two. It's foot the fraction and that's one of the seven multiplied by 3/14. Okay, now we're gonna go to step three. We simply gonna some multiply the top in the bottom. What's the negative times? A positive and negative one times three is three seven times 14. How good are you getting at your time? Sam will remember. You can use long division if you want, but you can see that seven times 10 is 70 and seven times four is 28. So 70 plus 28 is 98. Can we divide that spotted by three? It doesn't look like it. So that is good enough for on, sir, It's try to and 18 Multiply by through negative three. Enter five. We're gonna first Goddio put it over common. I mean, put it into improper fractions. Have a 58 times to 16 plus one is sailing team five times three is 15 plus 2 17 That's nice now 17 multiplied by 17 and 18 multiplied by five. That was our Step One, and now we're gonna simply move on to step two, because modification How we gonna do 17/17. This do long multiplication in 17 multiplied by 17 seven times seven is 49 times full, seven times Wyness seven past four is 11 and a magic 01 time seven a seven and one times one is one, and that is equal to nine plus eight plus 2 289 So this is going to now go we hear now 289 . Is it a negative or positive? It's a negative, divided by eight times five is 14 and that is good enough for your answer. Let's try some more examples. This last section negative two in the fifth, divided by negative three and four or five. So your first converting proper. We can see that it's a divide by negative. That's a negative over five 35 times two is 10 plus one is 11 5 times to his 15 plus four is 19. What is the next thing we doing? Multiplication. I mean, in division. We're gonna convert it to multiple. So we gotta flip the fraction where we make a times and then we say 5/19. Let's not forget I'm negative now. What's a negative times? A negative? That's positive. So I was gonna equal positive. 11 times five is 55 five times 119. What is five times 20? That's 100 minus. Fire is 95. Can we simplify 95? Can we simplify this further? We can divide both by five to get 11 and 19 and that looks about right. Answer. You might have noticed he has something very unique to multiplication. If there's the same name on top of the bottom, we can simply cross them out. And look what our answer is. 11 over here, divided by 19 and it's a positive. Let's have a look at this example 10.5 minus 15 sevens to make it to multiplied by negative something of a seven two times 10 is 20 plus one is 21 seven plus five is 13 or 12. So we haven't multiplication. So we gotta something normal to buy the top of the bottom. What is 21 multiplied by 12? Remember, just use long division. If you need two times is to two times was four and a magic 01 and two. What is this gonna equal to five to get? Keep practicing this until you can do very quickly. Said 21 times 12 is 252. That's a two over 14 is as a positive or negative. It is a negative. Can we simplify this further? Yes, we can. That is equal to negative. We could half both the top and the bottom. Ah, half of 152 is 126. Oh, sorry, this run. But it is 126 half of 14 7 and that should be good enough for Island said. Let's try one last example for 1/2 divided by 4 to 11 we make it first common at the bottom 11. These are both positive, which is nice and easy. Three times forward eight because one is nine. Four times 11 is 44 plus two is 44 pastors, 46 serene now division. So we gotta flip it. So that's gonna be nine over to multiplied by 11/46. Practice makes perfect. That's why we do a lot of examples. Nine times 11 is 99 and two times 46 is 90 to, and that is as far as you can. Simplify it. Well done, guys. That was multiplication and dividing. Remember, mixing and properties over this on next one is square and cube breeze. Practice mixing and proper. If you were struggling to convert that but otherwise, well, then great, it's 30. Common Fractions (5 of 8) Finding Square and Cube Roots: Okay, great. Let's we still uncommon fractions, and now we're going to do the square and cube roots of common fractions. This is quite an interesting one when we look at how similar it waas to square and cube roots were previously done on the similar laws. So these to link up quite nicely. Let's have a look at us in the exponents section. We previously learned how to find the square cube roots. So remember the square root of 25 that is simply quit. Five. Because we know that five multiplied by five is equal to 25 that's what that symbol is showing us. The square root of 100 is equal to 10 because 10 times 10 is equal to 100. So that was square roots. We also learned by Cube roots, another's Cuba. It's where why, what times itself three times or equal 125? And that is also equal to five because five multiplied by five. We know that's 25 multiplied by. Another five is 125 and that's what a cube root is. Where it's got a three at the top. The three means multiplied three times All the Cuba to 27 is 33 times three is mind. Times three is 27 right in exponents section. We also learned about the power laws and one law in particular that said, a number of a number to the power vein is equal to that number to the power of in divided by that number to the Parvin. Make sure you have watched exponents section, if you back in with us. So, for example, X over y over three is the same as X three over Why three. All four over to the bar three is the same as for cute over too cute, which we can take that further. Four. Cute we know 64 and too cute is eight. We can simplify the further t Kuwait. We could have even done that inside the bracket to start with. So how do we find us? Find the square roots and cube roots in a fraction. Something I want you to know is that the square root of 25 is the same as 25 to 1/2. That's exactly the same. It's just how we donated differently. Similarly, the Cubans of 125 is the same story as 125 to the power of food. That's that should be a one there, 125 to the party food, and similarly, you could do this for full. So the fourth root of 25 is the same as 25 divided by one of her four. And as you can see, there's a pattern here where that is the same as X over one that is the same as X and one of the three, and it goes on. So this could be this cube roots of any number. Eggs to the power of in is the same as that number X divided by one over in. It's the exact same. So we have our law, where that I just showed you accept a white and Parvin is equal to X in, over white in. So what if we did nine of a four to the power of 1/2 by this law is just write it differently by this law, or we can say that each nine owner half and for the hearth is the same. We can do that that you learned in your powerful and now we can see that Ah, half is the same as a route. So we can say that that's the square root of nine. I was a square to full and you can see that that equals 3/2. Similarly, we could write that square root, not four overnight. We switches around this time because that is the exact same. Is that or rather, that over there does having a square root over the entire thing? So that is the same as something riding the square to four of the square of nine. And that is simply writing to over three as you could. Easy work ups. Okay, The best way to do this is with some examples. What's the square root of 25 divided by the square root of 16? That is the same as saying this greater to 25 divided by the square root of 16 and that is equal to you know, this great of 25 5 is greater. 16 is full. What about the cube root of 27 divided by eight? Well, that is exactly the same as saying the Cuban of 27 divided by the cube root of AIDS, and you can easily solve that to be. This Cuban of 27 is three and a Cuban ages two, and that is your answer. This was that Now we have the square root of 64. We have the square root of this credit of 16 divided by 16 so we've got to start in the middle. So we keep that a square. What's the square root of 16 that is equal to fall? I was 16 and now confined the square root of each of those individual where the square root of fall divided by the square root of 16. And that's easy. You know, that is to over four, which is the same as 1/2. Well done. Can I have a look at another example? What is the square root of the cube root of 64 over the square root of 25? Don't get worried. You just started a little and then go out. What is the Cuban of 64? We know that that is 44 times for 16 times 4 64 What is the square root of 25? That's five. Now we do the both the same to the top in the bottom. So that is the square root of four over the square root of five. What's the square into four that is, too. What's the square root of five? You can't work out the square to five in your head. You need a calculator, so it is okay to simply leave this as the square root of five. And that can be your answer. The square root of a name. So that is OK, this is good. That's fine. Well done. Okay, What about the square root of the cube root of 64 over 125 multiplied by the square of 49/7 . Again, this is easy to do. We simply say the cube root off 64 over the cube root of 25 multiplied by the square root of 49 over the square root of 89 81. What is a cube root of 64? We know that's for what is a Cuban of 125 that is five multiplied by 49 the square to 49 a seven, and this crater to anyone is nine. So if you have a few batting with these. Remember to practice your timetable and watch the video on Whole Numbers Square or exponents, Square Roots and Cube Bruce. What is four times seven? That is E. That is equal to 28 what's five times nine? That's 45 that is good enough. Let's have a look at one last example. What's a cubit of 27 of the Cuban one? What's a Cuba to anything we know? That's three. What's the Cuban? A one If I said one multiplied by one multiplied by one, I get one. If I said the seventh root of one that is also equal to one. So that is equal to one, which gives our whole answer three. And that is that well done. That was square. It's in Cuba. It's next gonna refund. A percentage is 31. Common Fractions (6 of 8) Finding percentages of whole numbers: Okay, Great. It's now we doing percentages were still in the Common Fraction section, So a decimal percentage infraction off the same way of representing a number. You might have seen this already. For example, we write 1/2 in as many ways as possible. Ah, half could equals 0.5 get equal 50%. It could equal 50 over 100 equal 5/2 or killed because 17 over for as long as the number on the top is half the number on the bottom. So we could write double in as many ways as possible. That's too. We could drive 2.0 2200% or 200 over 100 20/10. 70/35 as long as the number of the top is double the number on the bottom. So how do we convert numbers or fractions to percentages? Okay, so how do we solve a percentage? This is the way to show an amount or share off a whole number. For example, what is 1/10 share of 100? So if we divided by 100 into 10 we can see this 10 tens. So what is 10% share of 101 10th is 10% share of 100 that is equal to 10. What is 30% share? 100. So we divide us, entertain and we 100 we said we want 30 of it and that's that and that is equal to 30. Similarly, what is 10% share? 50. We could divide 50 by 100 then times it by 10 is equal to five. So 1% means one per 100 10% means 10 per 127% for example, means 27 per 100. A nice way to think of percent is per 100. So think of the PR is the per and ST a century per 100. So 45 names 45 per 145% 45% off 100 is easy to do that's equal to 45 45% off. 200 is the same as 4500 so we can see this to hundreds there, so it's 45 per each of the hundreds on that is equal to 19 so x percent is equal to x over 100 or X over 100 is equal to X Person. So, for example, 25% is equal to 25 over 100. And now we can simplify to one over full. Okay, so if I said 32.75% that is equal to 32.75 over 100 time, could we go into that? In other videos? Let's have a look until examples 37 over 100 equals 37%. 12 over 200. That's the same as six over 100 which is the equivalent of 6%. 20 over 160 is the same as two of one in a six year. We can cancel the zeros, I mean over 16 which is same as one of AIDS, and you might have to put this into a calculated to work it out. But we simply have 0.1 to 5. They always to work the sum of, um, not using a calculator. I don't think you won't need to worry too much about that right now, but it it includes long multiplication and long division receiving a working off for fun. And that's 12.5% So 98.5%. Same as 98.5. Over 100. Okay. It shouldn't mean equals there. That's the end. So what is 5100? That is the same. We can cancel those out. Five of the 10 which is the equivalent of one of it too, which is equivalent of 50%. What about 7/84 7/84. If we divide 84 by seven equals 12 1/12. And again, you can put that in the Coquelin and get 0.833 And then it's 8.33%. Can. What is 15 Divided by 10. 15. Divided by 10 is the same as 3/2. Hm. What is happening yet? 3/2 is the same as one and one. Hoff. We know that one is 100% and we know that half is 50%. So that is equal to 150%. That's also you can What about 25.5? We know that is 25 0.5. Over 100. Okay. Oh, has taken this further. What you can do if you want. You can say that you can add the decimal this way. And if you add the decimal this way, you've gotta add 10 0 at the bottom. Same with this. We could say that's 255 over 1000. And that's another way you can solve the decimals. So Well, then that is our video. Impotent percentages were gonna go toe adjusting at by percentage, which is even more build up on this video before we get into work problems well under day. 32. Common Fractions (7 of 8) Increase or decrease a number by a percentage: Okay. Great dates. Now we're gonna be adjusting a number by percentage, this little common fractions section. Basically, what we're gonna be doing with those is increasing or decreasing a number by a given percentage. Right? The best way to learn this is through examples and nice practical examples. So let's get started. I have 70 apples. What is 50% off my apples. So 50% 70 times 50% is the same. A 70 time 015 on. You know that 0.5 is the same as 1/2. So 70 times. 1/2. And you can quite easily told that 1/2 of 75 of 70 is 35. Let's have a look at another example. I have 70 apples. How many will? I haven't total if I increase that number by 50%. Okay. You should have really beginning some idea off. We know what 50% of 70 is, right? So that's 70 times 50% is 35 and we want to increase the number by 50%. So we say 70 plus 35. And that is equal to 105. So another way we could do this it's a 70 times 150%. Now. The 100% is the 70 itself because we know that 7100% of 70 is 70. So we can just write adding another 50% here, and that is gonna equal 105 as well. So this is the same A 70 times 1.5. And that is 70 times 3/2. Because we say two times one is two pass one is three all over to you know how to do this. Make sure you watch the video and compote converting mixed numbers to improper fractions. If you don't know how to do that and then 70 we multiply the top and multiply the bottom. 70 times three is Turner and 10 and one times two is two, and that is equal to 105. Okay, let's have a look at another example. I want to increase my savings by 30% this year. I currently have 870 round. What will I have at the end of the year? Okay, so let's go the long way around. We can go 870 tons 30%. That is the same as, you know, percentage is that number per 100. So 30 per 100. So it turns 70 times 30 over 100. That is the science Samos 18 70/1 Time Street over 10. Because, you know, we can cancel zeros on the top and the bottom starting from the right, and that is equal to 2610 divided by 10. That is what would happen if you multiply the top. You get tournaments. 2610. Use a calculator or long multiplication if you need. And that is quite easy, because we can just cancel the zeros again and we get 261 now. We've gotta add 872. This number here. So this number here we knew his 261 so that's quite easy. 870 times. Oh, plus 261. And that is equal to 1130 rand. And that's how much rand we wanna have it the end of the year. But what would be the easier way to do this? So what we could do because they're 870 plus 817 times 30%. We can drop the brackets. That's all we've done there. Now we take out the common factor of 870. So 870 is the same as or this whole equation is the same as eight and 17 times, one plus 0.3. We know that 30% is 0.30 and we know that's a three. And we know that, Um, So if we times this out, we'll get that exact thing. So look, if we said eight and 70 times one that equals 18 70 18 70 times 0.3, that is equal to this part of the equation here. So that is the same eight or 70. We add those two together to get 1.3 and then we consist. Simply say that is the exact same. If you put that in, your calculator has 1131 and that is the easier way. So this is method One, and this is method to either will get you the same Lancet do what works the best for you and what you understand the best. I always encourage taking the longer way if you're struggling to understand that your first few times Okay, let's have a look at another example. I used to have 30 apples on now have 45 apples by what percent have increased my apples. So it's not the same question but asked slightly differently. So we go 13 times some percent, and that's going to give us 45. So what? How do we solve for this part of the equation? Make sure you watch the videos on Ulta Great and the equations if you're struggling to figure this out. But what we can do is what we do to one side. We have to do to the other. So what if three times the left hand side by one of the 30 that means would have two times the right hand side by 1/30? And what happens? Yes, 30 divided by 13. Because that is the same as 30/30. They cancel each other out because this is over one. So those cancel each other arts or gives 1/1, and then over here we have the question mark percent is equal to 45/30 because that's the same as 45/1 and a 45 times one is 45 1 times 30 is 13. We can simplify this further to give us 9/6 and even further to give us 3/2. What is three over to that is an improper fraction. So we need to convert it to a mixed number. And that is 1.5. And that is the same as one from 511.5. And that's as we know. It is 150% because we can take we add some zeros here. That's OK. And we take a decimal to across to get 150% over there. Okay, let's have a look at another example. I used to have a choose. I now have six shoes. By what percent have I decreased my shoes? So what percent have I decreased my shoes? Okay, let's have a look at this. You've got to look at the wording carefully. Year. This is gonna lead us in time. Next video onward Problems eight times as 30% is equal to six. So we followed the exact same mated we times by one of eight on the sign of one of eight in the side. Those cancel out nicely. So we left with one times the percent of a six of eight on the side. And that is the same as three of a full. So the percentage, or its server for the percentage of 0.75 or the decimal and the percentage is 75%. But be careful here. That's not the answer. I have 75% the mart of shoes I used to have. I haven't decreased my minor shoes by 75%. I have 75% of what are you serve. Therefore, I have decreased my number of shoes by 25% because we simply went 100 minus 75% to give us 25% and one last example. I have 200 oranges. I give 23% of them away. How many do I have left? Okay. How are we gonna do this? We give 23% of our oranges away, so we gonna be left with 100 minus 23% afterwards. So we start. We gave 23% away. So we're gonna left with 100% minus 23% and that is equal to quite easily 77%. So that is how many were left with. So we need a sticker out. What? 77% of 200 oranges is. So what do we do that we go to hundreds? 200? Sorry, that's Ah, just a zero two hundreds multiplied by 77%. So that is the same as 200 equals here, multiplied by 0.77 So you could work this out quite easy on a calculator. Something else you could spot you is that 200 is the same as 21 hundreds. It's quite easy to find 77% of 100 correct. So this is the same as 100. This is just for your interest times 77 0.77% plus 100 time, 0.77% for 0.77 And that is equal to 77 plus 77. This is just to practice working out things like this and that you can see is we know 70 plus 70 is 114. That's another 14 is 154. And that is how many oranges we have left. Okay, so we just finished the sectional adjusting a number by percentage. Well done, Great AIDS. We're gonna now move on to word problems, which is very similar to what we've been doing now, but including everything over yet, okay. 33. Common Fractions (8 of 8) Word problems: Okay, Great. We're not gonna go through some word problems this out. Last section in common fractions and fractions and percentages, they always will. Most often, you'll find them in word problems. Let's go through six examples to understand word problems. Better. You have 30. Both. Sorry. You have 300 ackles. 20% of them are bad. How many bad apples do you have? Okay, so let's first figure out what, 20% off? 300 hours, cause, you know, we know we're gonna have less than 300 good apples, so we're gonna go 300 multiplied by 20% rise. We know that that is equal to three. Hundreds multiplied by zero 0.2 because we moved the decimal from there across to. That's what we do to get into percentage. And what is 0.20 point two is the same as 1/5. So that's 300 multiplied by 1/5. And what is that? Equals 300 divided by five. 60. So we have 60 bad ackles. So how many bad apples we have? 60. Quite easy. How many good ones do you have? We have 300 minus 60 Good ones, which is 240. Let's have a look at another example. You have 500 apples. 50 are bad. What percentage of bad apples do you have? Okay, we now have to work the same question. The backwards the other wear owns. So we know that we have 500 ackles in total. Now, a certain percent of these Ackles we're gonna is just say question Mark percents, even though it would be better to use something like in eggs is equal to 50. So we know that 500 times a certain percent is equal to 50 bad apples. So how can we figure this out? Remember, we gonna times one of the 500 on both sides to get our question lock out to get our question mug out So we multiply by one of the 500. That's I, and one of the 500 on this side. So these are gonna cancel out. We're gonna have a question mark percent left on the sides, and that is equal 50 over 500. And you can probably quite easily see those cancel and 5/50 is equal to 1/10. So the question mark is equal to 0.1, but we wanted in a percent, so he times it by 100. And that is equal to 10% of Ackles of that. Now, the next two or the next example, all the examples from now I want you to pause the video and try them yourself. And then watch how we work. Walk through it. You see a bicycle being sold at a 25% reduced price. The price was originally 1400 rounds. How much is it? No. Okay, the first thing we're gonna do, say, 1400 rounds multiplied by 25%. What do you know? 2025% of you know 25 is one of a full. So 1400 multiplied by 1/4. So this is the same as saying 1400 divided by full, which I'm sure you could do quite easy. But why don't we start by dividing both by two? And that equals 700. Divided by two. What is 700 divided by 2 350 over one, which is equal to 315. So that is what? 25% of 1400. And is so how much is the personal? It is that less than the 1400. So the current price of the new price is 1400 rand minus 350 rounds. And you can probably quite easy workout that. That is gonna be 1000 and 50 round. Well done. If you pause that and got that onto yourself, try and pause the next question that we have here and see if you can get the answer again. You see a bicycle being sold for 500 which is 60% of its original plan price. How much was the original price? So now we're looking at the reduced price, For example, In this case, that would be that amount, so we wouldn't know. It has been reduced 60%. What was the original price? Okay, so we're gonna have the original prize. I'm decor. That o P original prize multiplies by 60% has equal 500 rands. Okay, That makes soon's There was a certain price could have been that 1400 multiplied by percent is equal to what the 500 rounds. So we're gonna have our original prize. What is the same as percentage as 60%. It's the same, A 0.6. And that is equal to 500. Now, how do we get rid of the Sierra Point? Six year? We're gonna multiply both sides by one of his 0.6. We're gonna most by this side by 1/0 0.6. So these gonna cancel out. And now we're gonna have the original prize is equal to five hundreds, multiplies by 1/0 0.6. How can we work this out? You could put that in your calculator and figure it out. What is 0.6? Now, remember, we can do something with multiplying and dividing fractions. We could say that that is the same as 500. Divided by zero point 6/1. Do you remember how If we change the multiplying and dividing, we have two swept the flip. The fraction. Remember those questions if you haven't, if you don't know how to do that, make sure you watch dividing and multiplying of fractions. There's a video a couple back in the series. And what is your 0.6 0.6 is the same. A six of 10. So that is the same as 500. I'm gonna move this over here. 500 divided by 6/10. Okay, now what if we wanted to flip it again? We could say that's five hundreds multiplied by 10 over six. Okay, it's work this out. What is 500? Remember this? 500 over one multiplied by 10. That's 5000 divided by six. Can that you might find the vehicle sellers to fight about two. First to get 2500 divided by three. Can we divide 2500 by three. You might need to use your calculator here. We don't get into decimal fractions yet, so simply go in your calculator. 2500 divided by three. And we're gonna get 833 rounds. And that is our answer. 800 aunt 33 ran on 33 cents. If you want to put it in, which is fine. We might have asked you for the states, and that is on. So Okay, let's move on to two more or two last examples. Okay, let's try last two examples. Remember, pause it and try it by yourself and then watch the working out. You buy two bags of sweets. Each bag is 50 sweets. The first bag is 30% rate sweets and the second bags 40% road sweets. How many red sweets are there in total, but can't. So let's look at bag one. It's a bag. One has 50 sweets multiplied by 30%. What do you know? 30% is the Samos 50 multiplied by 0.3 That is the same as 50 multiplied by 3/10. Would you can? Quite easy. Workout is 50 tons. Three is 115 divided by 10 and that is equal to 15. Sweets rate sweets in bag one. What about bag number two? Hopes back to There are 40% red sweets and back to so we say 15 multiplied by 40% and that is the same as 50 multiplied by 0.4. And that is the same as 50 multiplied by four over 10. What's 50 times for that is equal to 200 200 divided by 10 and you can quite easily see that that is 20 suites So what is the total monastery to have that is equal to 15 plus 20 but you can quite easy. Workout is 35 red sweets in total. See if you can figure out this equation or this question in another way, where you add the sweets together first. If you If you really cruising through this, let's have a look at the last example. You have a bag off. You have a sweet bag with 80 sweets. You give two of your friends 20% each. How many sweets and you left with? OK, let's first figure out what 20% of 80 sweets is. So 80 sweets multiplied by 20%. That is the same as 80 multiplied by 0.2, and that is the same as 80 multiplied by 2/10. What is 80 times to that is 160. What it remained. That's over 11 times. Time is 10 and that is equal to 16 each. You get each of your sweets friends, 16 sweets so you have two friends that you each giving 20%. So you're giving them 16 multiplied by two friends, and that is equal to 32 suites in total, in total. So how many do you have left? You obviously have a T minus 32 suites, and that is quite easy. You can work that out. That's 48 sweets. You must write what the answer is because it's a word problem. You have to rights what you're talking about at the end. So any fully ate sweets? Lift what? For your own interest. Tried at home. See if you can work out what is You give it your friends. 20%. So you're gonna have 60% left. So see if you can work out. I'm just gonna write us like this. 80% or 80 multiplied by 60% See if you can work. What that out is you should find it is very similar. Or exactly the same to that answer over there. Well, then radiates. You really do. Well, we've just finished the word problem section in common fractions. We're not gonna move on to decimal fractions. We get a little bit more exciting. And last year generated proportion and financial mess, which are two loads of fun sections. Well done. 34. Decimal Fractions (1 of 8) Ordering and Comparing: Okay, great. Let's now we doing decimal fractions and the first section in this section we gonna know the first video in the section is ordering and comparing off decimal fractions. Now there's more. Fractions can be confusing because of the fact that it's quite difficult to tell which is the biggest in, which is the smallest. And it's difficult to understand the size of a decimal fraction. And that's why we're gonna do ordering in comparing of decimal fractions first, let's go straight into it. Like I said, it can be tricky. But don't worry. We will understand by the end of this video, so we could have a desk more like through 0.3 to remember decimals. Anything with that dot in it, we could have 0.302 or we could have 0.3002 So if I ask you which of these decimal fractions is larger, No. Could you answer this for me? See if you can figure it out yourself, right. Be careful, even though this number here, this looks bigger than this number here, This, in fact, is the biggest number year, 0.32 0.30 to is in fact the second biggest. And this is the smallest out of these three numbers. So why's is the case? Well, we must remain, but that this is not 3002. This is not 300 to, and this is not 32. We could add. Zero is going on here forever rise. And if we added, zero is going on forever, we can see that this is 32 forever. This is 302 forever. This is 3002 forever. So if we were just keep it at four zeros, we can see that That's 3002. We can see that. That is 3000 and 20 and we can see that that is 3200. Because this is our 10th. This is Ah, 100th this off, thousands all the way down. So let's look at this. We're gonna get into that further, which is larger out of 0.10 point 06 or 0.54 Let's add the zeros. You see, this one has four. So let's have four zeros or four numbers in total. We just have to add one day to make its for on. We're just gonna add another three year. So what number is that? That's 1000. That's 600. And that's 54. So, quite clearly, this is the biggest number. This is our second biggest number, and that is our third biggest number. Because this doesn't have any tense. All hundreds, It only has thousands and 10 thousands. This one has 1/10 so it is immediately bigger. Okay, let's go. Which is larger 0.1 now that is equal to 1/10. Remember this here is our 10th. The closest went to the decimal in the right. Inside, 0.1 This is one hundreds. The 1st 1 is 10. The next one is hundreds. So this is 0/10 and 100. What about Sierra 0.1? This is one thousands. You can probably imagine what the next one is gonna be. 1 10,100 thousands and ultimately won millions. And this would go on forever. And you can tell that 1/100 is smaller than 1/10 remembers tense 100th South thousands. So if we looked at the number 7.123 that is the same was saying, saving over one tai O plus 1/10 plus two hundreds. Remember? Because at the bottom we says over there, plus three thousands. Now we could make these all over the same common denominator. You already know how to do that. So if we wanted to get them all under 1000 we would need to multiply this by 10 toe at Taney on the bottom, multiplied by 10 on the top were to multiply this by 100 on the bottom, multiplied by 100 on the top and multiply just by 1000 and most by just by 1000 on the top . So what did you get? We would get 7000 over one thousands, 100 over 1200 over 1000 and 300,000. And what is the number we gotta get? 700 plus $100 21st 3 7123 over one thousands. Okay, so this is the equivalent. Remember, we're comparing it as well. To that. It's written the exact saying there. Okay, let's get into some examples. What is larger? 16.209 or 16.210 Okay, let's start. 16/1 has to over 10 0 over 100 cars. Nine of 1000. That's what that is over there. 0 100 to hunt, 10th 16 over one. So we make that all over 1003 have to time that by 1005 100 that by 10 which stayed a zero and that by nothing. And then we got the number. We addle the top 16,209. Okay, what about the number? The top here, That's 16/1 to over 10 1 of the 100 0 over thousands. So we go, we add all those together, six animal room. Together we get 16,200 Tim and you can quite easily see now that they're both over the same common denominator that this number is one bigger than that number. So this is clearly the bigger number. And then we do our cocked out eating the bigger number. So let's look at numbers without the decimals. First the left off the decimal, then the right of the decimal hot. What? I mean by this right to compare whether numbers bigger. First, look at the left of the dismal. We see seven points. 986 is a smaller than or bigger than 8.2. We don't even need to worry about the decimals because we can clearly see that eight is bigger than seven. So we put our crocodile eating eight, and we say 7.892 is less than 8.2. But what if they are the same on the left hand side of the dismal? So this is the left hand side of the day, Small. Here's our They're small. So that was obvious. Now we're looking on the right inside the dismal because they are the same number. So now we see. Okay, let's start here. All right, so let's draw this line here. We start here and we're going that way. We start here, and we're going that way on. We're gonna keep going until we find one number bigger than the other. Full is clearly smaller than six, so six is bigger, so we're gonna make our crocodile that way. 5.4869 is less than 5.62 So this is the same as 5.62 5.4869 Remember, we line up the decimals and we could add two zeros there if we wanted What about 9.99999 and 10. We don't even need to look at the decimals. 10 is clearly bigger than known. No matter how many decimals we add there, what about 5.987 and 5.98? Which one here is bigger? Okay, so when we can see that everything on the left hand side of the desk was the same So we've got a now look on the right hand side of the day. So also, we start this way and we got to go to the right. We can see the nines day the same. The IDs, they're the same. The seven. What is here? There's a zero year we can add a zero. Remember, Seven is bigger than zero. So this number is bigger than that number, and then we're gonna say 5.9 and seven is greater than 5.98 Now, remember, we only concerned with what's on the right hand side of the decibel. Let's have a look at some more examples what about 10.999 and 10.9? Right? They're both the same on the left, so we have to look to the right of the decimal. So that's 10.9, and that is 000 room and at his ministry once. That's 99 so there is the same. That's a nine and thats zero. So we know that that is bigger. You're probably getting these quite easy. No. What about 7.1 and 7.10? Right? We can see that's the same zeros of the same. This is the era and is a one day, so this must be bigger. What about 10.9 and 10.1 right there. The same. So we go start on the right of the decimal or Goethe to the right of the decimal. We got a zero and a one. Ah, the one is bigger. So no matter what that is begun number. So 10.0 is 999 is less than 10.1 I'm 10.1. What about 32.23 and 32.32 Again all you need to do is look at the right of the decimal, and you can see that that three is bigger than that, too. So we can say that is bigger. 11 points there is there nine and 10. We don't even need to look at the decimals. That's an easy one. And what about 2.93 and 2.3 again and easy one? You started the rights and you go down the decimals and you can see that two countries larger. Remember, we can only compare each of these lines with the other one, so there's no point you couldn't compare the zero with the other zeros here. It has to be in the 10th in the hundreds or the 10th the hundreds and the thousands or the one over. That's how we says, Okay, could one of attending your calculator one of 100 year Coquelin and see what you get your notice. How one of a teen is bigger than one over 101,000. Let's have a look at some last page of examples were arranged from smallest to the largest rides, 10.99 10.9 10.99 10.1. 10.9. Okay, we can see that all tens, So that's doesn't help. So we gotta started the decimal, and we're gonna go, right? We're going to go, right. We're going to go rights all at the decimals. Okay, firstly, we see that there's a nine. So they 90 there's a one and a nine, so these we can all ignore, but here we can see a zero, and then a one. So smallest is gonna be that 10.999 Next one is going to be 10.1. Cost that writer We checked the first decimal place. Now let's check the next one. There's a nine again there. This is zero here, and there's a to hear. Ah, we know that zero is the smallest, so that is gonna be 10 points. Nine can cost that whole enough. Then we got a to 10.9 to, and then we got this big one here. 10 0.99 99 Okay, let's arrange these from greatest to least you can see have used different terminology here . There's 7.17 point 107.17 point 117.901 and 17.1 What is the greatest? We gotta seven everywhere but a 17 over here. So we can immediately cross that 17 out because we know that that is gonna be the biggest one year. So what's next? Could seize all the same. We gotta go along from the decimal. So we got a zero there. A zero they remember We're looking for the biggest one. We got a one here. We got a zero here and we got a zero here. OK, so it's clearly going to be the one, because that's the biggest. So the next is gonna be 7.1. Okay, so we've done our first zeros, our first decimals. Now, let's look at the next day. Small. Here. We have a nine year. We ever one here. We have a one, and here we have a zero. Okay, we can clearly see that the nine is the next biggest. So that s 7.901 and we can cross it out now. We have to ones. So we need to go to the next one. There's a zero and there's a one. So then we have. The one is clearly bigger than zero. So that is 7.11 and then 7.10 And then, lastly, there anyone left. Our smallest one is 7.1 and that's your answer world. Andre dates for getting through that video. It's very important to make sure you understand how to order and compare your decimals and what a decimal actually is. Care around the fitting. Your calculator. Make sure you understand it before we go to the next sections. Well done, everyone. 35. Decimal Fractions (2 of 8) Equivalent forms: I could credits. We're not gonna be doing equivalent forms. Now, this video you should find extremely easy if you understood the ordering and comparing video properly. So if you're better with equivalent forms, it means you need to go back and watch this video here que equivalent forms our decimals, fractions and percentages that are numbers that may look difference but are in fact the same. So listen what we see here their number that look difference. But the value off the number is the exact same. Okay, so for example, Sierra Quinn four is exactly the same. A 0.40 These are both decimals, right? Which is the section we're currently on on decimals, although they look exactly the same. But it can also be percentages and fractions, so four pertain. What it's for within is forever 10 a fraction or percentage. You know that for retainers, a fraction, and that's exactly the same. Although it looks different. That's exactly the same is 40 over 100 that's why we allowed to cancel those out. You can see that equals exact same, which is the same as to over five. What if you divide it each year by 24 Divide about two is foot two and 10. Divide about two years. Five and thats exact. Same is 40%. That's easy to know is or percentage okay, so you can add as many zeros to the right of a dismal, and it will be unchanged only more accurate. It's very important one day when you're doing fancy science and mass to know that although these are the exact same number, this number here is a lot more accurate on. We'll get into the detail of exactly what that means in the next couple of videos. But just know that these numbers are the exact same, just different forms. This number is, however, sightly different in the fact that it's more accurate. Um, but don't worry too much about that right now. Let's go through some examples. What is equivalent to 0.25 self first question 0.25 Is that the same? A 0.25 0.250 a quarter, or 0.25%. Okay, Is there going to five the same 0.25? No, of course it's not. We've got it started the decimal point and it's got to be the same from the decimal point. As you can see here, the very next point is too. So the very next point here zero No, that is not equivalent. What about 0.250 or I mean, we can add zeros of the NGO so we can see Yes, that is the same as 0.250 So yep, B is the same. What about 1/4 is 0.25 the same as 1/4 was your 0.25 is the same as 25 over 100. You can probably see we can divide both those by five to give us 5/20. And then we can divided by five again to give us one over for So yes, 1/4 is 0.25 What? About 0.25%. Now, don't get confused here. This is not the same. De is not because this is the same as 25%. That would be right. But unfortunately is not 0.25% because when we going from decimal 2% we've got to go to bounce two decimal points up. And make sure you have watched converting decimals to percentages. If you're betting with that, okay, let's have a look at another example. What is zero point saving equivalent to 0.7 0.14 7/1 or 70%? I think you can quite easily see that these are the exact same looking. Hence, there are the exact same value. 0.1 full. No, that can't be rides. Is that double 0.7? No, it does not double 0.7. What is 0.7 multiplied by two? Does that equal it? No, that is actually equal to 1.4 Nazi era 0.1 full. So this is not it anywhere. Even if it was that that is not the same. Answer is, you know, point saving the same A saving over one. We know that 0.7 is the same seven of a 10 but it quite clearly isn't the same as 7/1. So no, it's not. What about 70% out we can see the 7/10 is equal to 70%. Great. So that is definitely equivalent, right? Let's try 0.55 is 0.55 The same 110 of a 20.50 510 of 200 or 0.550 Okay, you can quite easily cancel. Be out. I'm sure after practice we can see that That's five. The second decimal was not the same as that zero not be. You can see that we can add a zero year to make that the exact same. So it must be be must be d What about these two? It's something 510 0 210 over to is the same as 55/1. If we divide both sides by two, which is equal to 55. Okay, what about 100 10 of a 200? If we divide both sides by two, we're gonna get 55 over 100. Okay? The 55 of the 100 is the same as 55%. And we know that 55% is the same. A 0.55 are that must be equivalent. And that obviously is not okay. The last example in the section 0.63 again. You can so easily see that that is 0.63 and must be the same. What a 6.3 6.3 no is clearly one decimal away. It is not 63 of one. Hundreds is the same as 63% so we know that those two equal and we can tell easy that is 0.63 So that must be an answer. And that must be an answer. If you struggle with that stage and make sure you watch the ordering and comparing If you struggled with some of the percentages, maybe you need to watch these videos. Refresh yourself with these or official knowledge with these two videos or percentages and adjusted by percentage, that should definitely get you in good steed for next couple of sections on decimal fractions. Well downgraded its 36. Decimal Fractions (3 of 8) Rounding off: okay. Credits we now doing rounding off off decimal fractions. Now again, rounding off, you need to have been able to complete these two videos and make sure you understand them correctly before you saw rounding off and we're gonna get into one. Is rounding off mean, how we gonna dirt and learn some rules? Okay, let's get started right away. Why do we round off decimal numbers? You're just gonna briefly talk about why we do this? Running off decimals always depends on how accurate we need something remaining. In previous videos, I said 0.4 is the same as Europe. When 40000 They're the same number. They mean the same. But the one is a lot more accurate. So all depends on how accurate we need something. So say you decide to run a 5.0 kilometer run. As you can see, we round this off to one decimal. That means is one dismal place here. If you ran a 5.4 k run and we round it down to five cares, there's no decimal there. It would make a big difference here. Your running 400 meters further than your five kilometers. That is a big part of a red, your rotten. So what if your run was 5.4 on four K's on? We wanted around it down to five cares. That would make a much smaller difference, as and that is only 400 meters. So as you can see, there's a big difference here between a five K run and a 5.0 Karen, because a 5.4 carry on could be the same as five. Kate rounded down. But a 5.4 K would be the same as that. This would be the longest that could be to round it down to that, and this would be the longest disco beat around it down to that. So it's all about how accurate we want it. So if you're running a five k run, you would hope that it's a 5.0 kilometer run, cause that shows that the most there could be further is 400 meters and is 14 meters. Okay, we're gonna get more into that. But this is essentially why we round decimals off. It's not going to how we round decimals off step one. We need to decide how many decimals we need around it Down, too. And then Step two, you're gonna figure out if you're gonna round it up or round it down. Right? So if I say the question says round off to two decimals, that's easy. You being asked you being told to round it off to two decimal places. What does that mean? 6.242 currently has three decimal places. That means there's three paces on the right hand side of the decimal. We want around it down to two. So now we're gonna decided we're running it up or down to is so close to zero. So we're going around it straight down to 6.2 full. So we run a Darwin we sent. Let's make that to a zero. And if there's a zero, they we don't need to bother writing it. So that's 6.2 to 40 and rounded down to to 6.24 And that is our answer. That is not an answer. That is our answer. One of about 15.418 round off to two decimals right now here eight is much closer to 10 then it is 20 So we're going around it, huh? Here we round it down. Now we're gonna round it up. That's it for this question is up. So how are we going to do that if we round eight up to 10? You know that when you're adding 10 you've got adizero. I mean, a one. You gotta carry 10. So that is the same as 15 0.4 to Because we had to add a Tim. You're getting this. Okay, We're gonna carry with some more examples. What if we round it off to one day? So what? I mean by that. Here we have two decimals, and we wanted to just have one. 4.71 one is so close to zero, we're going around it down. So that's gonna become 4.7. What, about 5.79 or we're gonna run that up or down. Nine is so close to 10. We're gonna round it up. If we add a 10 we're gonna have to carry a 10 to make a 5.8. Okay? How do we around those numbers? What about five? The decimal five we rounded off to one decimal. So the question is round off to one decimal 6.25 Now you tell me, is five closer to 10 or closer to zero? So this is difficult. Now we gotta think, is five closer to 10 or zero. Now, I'm gonna tell you the rule. Whenever we have a five, we round it up. Five always goes up, you guys. So this is equal to 6.3. Let's do another one. What about 15.85? What is five always do? It goes up. We wanted to one decimal 15.9 because we carried out tens. As you can see, we're getting less accurate. Ear. This is no longer as accurate. What about full? So do we Round four Oppa's? Well, no. We around four down. So 6.24 is gonna equal 6.2. As you can see, rounding down is always easier because you don't have to change the number before it. We're here. We go pass one class one. What about 15.84 were going around that down to get 15 points 8 to 1 decimal, remember? So when we have a five, we always go up. When we have a four, we always go down. That's what you got to remember from this video. Let's try some examples. Round off toe one day. Small. So currently there two decimals. We need to take it to one. What is four do full goes down. So that's gonna be six 0.0. Okay, now what? About 15.95? What way do we round? Fine. We run five up. So they're five goes to 10 so we would have a zero. That 10 but now nine also goes to 10. So, what is this gonna equal? Watch carefully. This is equal to 16.0. Did you guys see what I did there? 15.9. We rounded up from five. We rounded up to 10. What happens with the 10? We bring the one across toe, add the team here. So we add those two together. Then this all of a sudden goes to 10. So we're gonna take the one across to 15. And what is 15 plus one? That is equal to 16. Did you see what we did there? Okay, we're gonna practice a more round off 23 decimals. 1.234 What is that gonna equal? So we got four decimals. At the moment, we only want three. So that has been equal 1.2 The focus down. Making a three. That is very easy. What if I asked you around this down three days? Mel's OK, so don't get too worried about this. But as you can see, there's 3456 decimals. So we first got around it down to five decimals, four days miles and then three days. Mel's. So if we round this five off, that becomes a five because five goes up. Now we have a five year does five got up or down? Five goes up making that, too, does to go up or down to goes down. Keeping that a three. We could easily have seen that instead, one will never, ever go up to five, so we could have left that there and just gone through. But it's always best to work from the right to the lift, so we're gonna get that answer equals two. Not I'm sorry. 0.923 right? Let's try some more tricky examples. Round off to one decimal 0.4 That is equal to 0.0 zero. Because around that down and we got to get it to one, that is 0.0 is our answer. What about 15.745 Ok, so that is going to go. We first got around it. 2 to 15.75 and then we've got around that again to one, which makes it 15.8. Ah, well done. Grade eights round, off to zero decibels. What does that mean? 1.834 We can see that eight is gonna go up. We don't even need to worry about. Those two is not on the brink. It's not a four or five, so eight is gonna go back. What does that mean? 1.8 rounded to zero decimals. I don't see a decimal spot is equal to two without any decibels. There's not 2.0. That's just too. What about zero for nine again? That's gonna go up to one. And that is equal to one, not 1.0. Just one. Let's try another round of to one decimal. 0.4445 Now what should we do you? We wanted it. One decimal. This currently five decimals. Right? So we're gonna have to round that down. So that is gonna equals 0.0 for full. We got around this five up to make that of five. It's round that down again. 0.4 were around that four after because of the five. So that goes to fires. So we need then round this 40 point fire at Sorry. 0.5 Oh, you mean just scratch that out 0.5 and we round this up again. Two equals 0.1. And now we had one decimal. So do you see what happened? They we gotta be very careful. And that is equal to 0.1. Okay, great dates. What about negative 5.35? That is equal to negative five. We can see that. That five is gonna take that to a full but four still going dance that is equal to negative 5.0. Remember, you've got to keep your negative. It's run off to two decimals, 1.84 full. We could quite easily see that the four is going down to give us 1.84 0.2. How do we round that? Off to two decimals. We can only assume that that is 0.20 Well done. Grade eights. You really doing well? And that is our rounding off section. We're not gonna go into addition and subtraction, multiplication and dividing by decimals. We have touched on it a little bit. Oh, sorry. Here. Multiplication of decimals. And we got to get more into it now. 37. Decimal Fractions (4 of 8) Adding and Subtracting: Okay, Great. It's now we're gonna be doing adding and subtracting of decimal fractions. Now, you really need to concentrate here and make sure you don't get confused between adding subtracting and multiplying. Dividing. This is gonna be the next video after these two adding and subtracting in columns with decimals like you all remember how to add to whole numbers. You have 504 157 and 22. And you said that sevens and the two of the ones the two and this five of the tens and the four is the hundreds, right to your 405 10 7 ones you have to tens of two ones. And that was easy. You lined up all the ones you lined up all the tens. You lined up all the hundreds, even though there was zero hundreds here. And you add them together. You said seven first two was 95 for 7 to 7 and four per zeros form. That was your answer. Unless she could have switched the other way around. And you have got the exact same number because remember, you had a line. Everything So 2% is 92 plus 57 and 40.7. His folk sorry, zero plus four is for right. But what was mostly important here was that you're keeping the hundreds tens and ones and line. So those were ones. Your tens, your hundreds. Now what if we would add days? Miles 457.5 added to 22.5 So now this is your tents, These your tens and this is your hundreds rights. It's a bit weird to say, but you'll notice that there is in the word and that's because it's 1/10 and one over hundreds yet. But don't worry too much about that. We will get into that. And we have done that already. But what is the most important thing here? We've got to a line on numbers. So now we not only a line or one's are tens and hundreds. We also have to align our tents and ah, hundreds in this case. Now, as you can see here, it is much easier to just allowing your decimal. So we have a lineup decimal here, and that is a way to remember it. Always align your decimals. So what would we? Do you remember that? Could be a zero here. Just says there could be a zero here. You could fill in the zeros, and we gotta still started the rights we extend this across. 50 is five zero plus five is five. We've got to add a decimal in line. We've got to keep it there. If this winds over, we carry a 10 and act exactly as normal has come here to plus seven is nine to pass. 57 and zero plus four is full. And all answers 479.55 a que. Okay, let's go into the examples. We have 1 123.45 1st 343.81 Right? So remember, we gotta keep our decimals in line. This is easy. Causes the same amount of decibels. 145 has 343.81 five past one is six. Four plus eight is 12. So we're gonna add the two. Carry the one. Don't forget your decimal. Three plus three is six plus one is seven. Two plus four is six and one plus three is full and has 467.26 What about 123? That's 43.9 Can we add our columns? So let's go 123. Remember, that's the same as 0.0 And so we gotta make sure that that point is the same. So on this side is a three. And there's a full and on the side that 009 I can remember this also. Zero there. You don't need to write that. Zero plus zero is 90 So zero has nine is 9000 zero class here again is your Oh, keep our decimal in the exact same line. Three plus three is 64 plus 2610 is 1 166.9 What about 979.29 1st 20.7109 Okay, we get our columns, we start by riding the top number. 979.29 Remain minutes, decimals. We're gonna keep the same. So on this side is zero, then a two. And on this side, there's a 7109 So zero past nine is 9000 Nike was one is tens. We had a one on the top. Carry 10 one plus 23 plus seven is 10 again, so we carry the one. One past nine is tens. We got us here again. Whatever forgot him a decimal. We had a one. Then we have another 91 for seven, but is eight plus two is is 10. We got it at a day and nine has one is 10 and we can add up 10. And that is equal to 1000 0.9 Okay, make sure you keep track you on. You see, it's treated exactly like a addition and subtraction. The only difference here is the fact that you are keeping of decimals in line, which is exactly what you were doing with whole numbers anyway. So subtraction in columns is exactly the same. Let's go through some examples for 2416.0 to minus 543.27 I encourage you to pause the video and try it yourself and see if you get the same old. So So let's write this 2416.2 We keep the dismal to the same way of 27. On the size we go. 345 on the side. What is to minus seven? Ah, we can't do that. We're gonna have to borrow one from here. But there is not one day, so we're gonna have to borrow one from here. So, six, we're gonna turn into five zero. We're gonna turn into 10. So 10 we're gonna cross art and make it nine and Addis a negative 12. So we borrowed too across you. 12 minus. Saving is five, nine months to seven. Don't forget your decimal. Five minutes. Three is too. Don't get confused about the columns. One minus four. We can't do so. We're gonna have to borrow one from there. Make it a three. Make that 11. 11 months. Four is seven three months. Five clone to borrow one. Make it a 1 13 months. 58 and in one minus. Here is one 1872.57 It's trying of example 73 minus 23.7 So we got 73 Remember, there's a point day point day 23 0.7 Now we call minus anything here. They're zeros here, So zero minus seven. We can't do. We need to borrow when we can't borrow one there. We need to borrow one year to make that two. That becomes a 10. We cross out the team to make it a nine we borrowed by crossing it out to make a tonight. We made that A 10 10 minus seven is equal 39 minus zero is equal to nine. That made sense. 10 minutes. We have a point Day zero. So what we did here was 0.93 plus 0.7 is equal to one that makes sense. So you can see that answer drives tu minus three. We can't do. We gotta borrow seven to make it a six. And you know how to do those? 12 minus three is equal to 96 minus twos for And that is our answer. 49.93 Last example in the section 5235.567 minus 201.6208 Let's start 5235 Pause it and try it yourself. Undergo quickly through this. 567 Minus remain about decimals. Have a one a zero. Keep it in line at two. Here we have a six to a zero and eight. So we got we've got a zero there. Remember? We can add a zero seven or zero minus every conduce. We gotta borrow one to make that six out of 10 there. 10 miles. Eight is to six months. 066 months to his full five months. Six you climb. Do we gotta borrow one? So we make that a four. We make that 15 minus six is nine. Don't forget your decimal four minus one is 33 months. Heroes three to minus 20 and five minus heroes five. And that is 5000 and 33.9462 Well done. Great is you really do? Well, that was additional subtraction. We're gonna not put a video on multiplication with decimals before Device division with decimals 38. Decimal Fractions (5 of 8) Multiplying Decimals: Okay. Great dates. Now we doing multiplication with decimals. Okay, So, in addition, it was so important to line up those decimals. Keep them in line, keep the hundreds, keep the tens, keep the tents. It all had to be in line. Multiplication is slightly different. So said we wanted to figure out 27.23 multiplied by 14.5, so we we could simply do this. Now, here, you'll notice there's no correlation between the decimals. They're not in line. And that is okay. Even though using this method both will work. And I'm gonna show you all of that right now. But let's do this together and see what we have here. Now you notice there's two decimal places here. I'm just gonna right at the top there. Two decimals. I'm just gonna decimal and at the top at the bottom is one decibel. So in total, there's three decimal places. A k Let's not do this whole multiplication. Ignoring all decimals. What is five multiplied by three. You should not have to do this if you don't watch the video. Multiplication within columns five multiplied by three is 15 have a one five multiplied by two is 10 plus one is 11. We have a one five multiplied by seven is 35 plus one is 36. I have a six and a three five multiplied by 2 10 plus three is 13. We had our magic zero four multiplied by two or three years. 12 We had a one four multiplied by eight is a form of fiber to use a plus. One is nine four, multiplied by seven. Is 28 28 carry too four months. Blabber, too, is 8002 is 10. We add. Another 20 is now because you're on the hundreds column or the next list the next digit. The next. Right down one times three is three one times two is two one time seven A seven one times two is two. You only gotta worry about you comes here to make sure these are in the same column. Now what do we do? We do. In addition, five plus zero plus 051 plus two is 36 past nine is 15,000 other threes. 18. Carry one, 10 11 12 13 14 4 Carry one, and that's 12 plus seven is nine past three over there. Now we have this answer. What is what? What does it mean? Where the decimals Well, we can add the decimals. We can simply say there is a total of three decimals. So we have to start you and go three decimals back. And that is our answer. 394 point eight 35 So there's three decimals there. There's 123 Okay, now I want to try another example. What if we said 25 land time? 75%. You know that 75% of 0.75 So that is the same as 25.0 time. 0.75 What if we said 25 round 250.0 time. 75%. So we know that that is 25 round 250.0 times 75%. Now, here you can see we've lined the decimals up and gathers a decimal here, and we haven't lined it up. I'm gonna do these two ounces. What is five times five? That is five to what is five times 10. 12 plus two is 14 2 out of 14. We carry out zero What is seven times 5 35 5 times three What a seven times to 14 plus three x 17. What is your times? Five To add a magic zeroes. 0 to 500 times 20 So we add these all together we put a big plus of five over there. Nine over there. Eight over there and one over there. Now, how many decimal places did we go? We have to go to back and not so. Yeah, they're not decimals. And here, the two decimals. So in total, there are two decimals. So we gotta go 12 And the answer is 18 runs, 95 cents. Okay, let's do this one, though. What is seven times what is five times 00? What is five times there again is 05 25 25 five and we add up to five times 10 is 10 5 times to his temples to 12. 70. We had imagined 07 times, 00 times. Another 007 times five is 35 at five out of 37 times two is 14 plus three is 17 zero times . Here we add a magic zeros zero Time 000 Time. 000 times 500 times 20 We had a big plus year zero on the side. Zero here. Five year saving here. Eight year and one year. No, But now what I want to show you is here. We had two decimals. So you had his two decimals here. There's two decimals. What has the total decimals? There's four decimals in total. So what does this mean? This means that we need to go. Four decimals. Sorry, there was a one here. We need to go Four decimals back. 1234 And that is not 18 runs 0.75 0 I've realized Made a mistake. You What is five times five is 25. So we add our five were added to five times two is 10 plus two is only 12. So there's only 12 years. So 22 plus five is seven. And that was a seven over there that made that 18 75 on this 18 75.0 which you know is the same 18 rand and 75 cents. And that is the exact same answer. Well done. Good that your radio do well and you can see, though, how this modification and column works. Okay, so what is 57 round and 26 cents times 12.5%? We know that 12.5% of 0.1 to 5 so we can easily say that is times 57.26 and we put our 0.125 year. Now it's multiplied or like you know, these are not in line. That's fine. Five times six is 30. We put a zero in a 35 times. Two is only 10 plus threes. 13 Put out one year, five times seven is 35. We put out five. Here we add our three five times five is 25 past trees. 28. We had no magic zero. Never forget the magic. Zero. What is two times six is 12 plus a one of a year. Two times two is full. Two times seven is 14. Keep it in line. Remember these in line 14 at a one year two times five is 10 plus. A one is 11. Now we add to magic zeroes. Don't forget that one time six is equal to six. One times two is equal to to one time. Seven is equal to 71 times five people to find. Now we could add our magic zeros again. Three of them. But we can see that zero times. Anything is just going to eat well. Zero. So there's no point even writing that road, but I've done it to show you. 00000 23 plus two is five. Five past four is nine plus six is 15. Carry one eight plus four is 12 plus. One is 13 plus. Two is 15. Carry one seven plus one is eight plus. Two is 10 past one is 11 and then one past one is to pass 57 How many decimals do we go back? There's two decimals here. There's one or three decimals here. How many decimals are there? In total? There's total. There's five decimals. So we add those two together. So we go. 12345 and we get 7.1555 So we had seven rounds. 1555 Now this is money, so money we're gonna have to probably right round it up or down. We know that five goes up. Five was up, so that is the equivalent of seven rand and 16 cents. And that is 25. That is 12.5% off 57 around 26 cents. Walden Great AIDS. 39. Decimal Fractions (6 of 8) Dividing Decimals by whole numbers: Okay, Great. It's now we're gonna be doing dividing a decimal buyer desk rule. So in the last few years, we did dividing by whole numbers. But now I'm coming back to divide by decimal numbers. Okay? You've already divided Ismael by whole number. Now you want to divide a decimal by a decimal 0.5 divided by 765.6. Well, 765.6. Divided by 0.5. Now you already know how to divided this small by whole number. So how can we make this? Ah, whole number. Right. So we said 765 divided by 0.5. What if we divide? This is exactly the same of how we writing this out. 7 65.6 So he divided by 0.5. So now this is the same way to write this out. What if we multiply both the bottom in the top? By 10 we would get 7656.0, divided by 50.5 divided pertinent just goes one across. And that goes one across. Now what do we have here? We have a whole number. That's great. We also have a whole number here, but the desk more happens to be 0.0. So we're gonna move out decimal right up here. So what? We go 75 How many times says five going to salmon once. One times 55 seven minus five is to bring down the six times. Does five going to 26? That's five times five times five is 25 26 minus 25 is one. We bring down our five. How many times is five going to 15? Three times We keep our line. Very careful here. We brought that down to bring us down here. Three times five is 15. We know that 15 minus 15 is zero. However, we gotta bring down a six. How many times is 5 to 6? Just one time. One times five is five six minus five is one. We bring down our zero. How many times? A Five going to 10. That's twice and five times two is 10. And we know that T minus 10 is zero. And that is good enough. We could add a zero that exact same thing. So if you times that number by five, you'll find we get the exact same own, sir. 1531.2. So if we want to divide a decimal by decimal, we've got to convert into a whole number. And the way we do this is we take the side and we see how much we've got. A times it by and told becomes a whole number. In this case, it was 10. Let's try this example. 0.12 or sorry. 65.7, divided by 0.12 So the first thing we're gonna do is get it to a whole number. So here we have to divide by 100 remove it to a cross. So what we're gonna do, you gonna move this to a cross? So that's going to become 12 time a divided 606,000, 570. We come from there today. Today. That's two across 20.0. South Point is here. So we moved us to a cross. So how many times is 12 going to 65? We know that 12 times 5 60 stories, eh? 55 times 12 60 65 to minus 65. Bring down a seven. We know that that must be four. Four times 12 is 48 57 minus 48 is nine. We bring down a zero. How many times has 12 going to 90 12 times seven is 70 plus 14 and let's 84. So that sounds right. 77 times 12 is 84. So 84 minus 90 minus 84 is six. How many times is 12 minutes or so? You could do that. We gotta bring down zero. 60 100 times was 12 going to 65 times exactly in 500. Type was 60 and that is remaining zero. So 574.5. And that is our answer. Because we multiply this side by 100 we had to multiply the side by 100. So if you said 547.5 multiplied by 0.12 you would get 65.7. Let's try one last big example. 3.70 to 57 Divided by 0.32 We're gonna move this to a cross, so we're gonna have to move this to a cross. So this time you decimal over here This is all we are doing in long division with decimals . So 32 goes into 37 1 time one times 32 is 32 37. Minus 32 is five way. Bring down as your own. How many times it's difficult to going to 51 time One time Study Tuesday to 50 minus 52 It's quite easy. We can see that That is 18 we bring down up to remember Got out decimal here. How many times is 32 going to 18? Okay, what is 32 times fine So we say 32 multiplied by five. That is equal to 160. We can see that That is gonna work. So we say 0.5 five times 32 is 116 1 80 to minus 1 60 is equal to 22. 22 We bring down our five This getting quite Bigas you can see 220 So we say What is 32 multiplied by Saving is trying 210 plus 14 is 224. So that's too big. What about 32? Multiplied by six, 118 plus 12 192 that will do 192. Well, that's a 22. So we say that six times six times 32 is 192. We said to 20 minus 1 92 We can quite easily see that that is 28th. We bring down our seven. How many times this try? 8 32 multiplied by eight, and that is equal to eight times three of eight times. 30 is 214. Plus 16 is 256 256. That looks about rights. So we say eight. Eight times 32 is Turner 56. What is 287 minus 256? We know that that is gonna equal. There's four there. So that's 60 Placid 27. Sorry. Four plus 27. It's 31 US. We're getting close now. We could add zero if we wanted. We bring that zero all the way down 310. We know that's gonna be the nine nine times pretty too. It's so if we go 32 times nine, it is equal to nine times studies. 30 is 270 plus 18 is 288 288. So 288 310 months. 288th and that is gonna equal. We have 20 day 22 this is going to go on for a while and we could leave. That is our answer. 32 in between 20 and that will be on 11.56890 And it could gone. You could ask. That is just two decimal places in which cast onto equals 11.57 Remember, because that's an eight. We round that up. If it was two decimal places, be careful to read what the question says most the time. It was in quite nicely like those. Well, then radiates that was working on the vision. Now we're gonna go into the last video, which is square roots and cube roots off this decimal fractions section 40. Decimal Fractions (7 of 8) Dividing Decimals: Okay, Great. It's now we're gonna be doing dividing by whole numbers. Right? So we've done Long division where we said, for example, 765. Divided by five. We said, How many times has five going to seven? We know that's 11 times five is five seven. Minus five is, too. How many? We bring down the six over here to make that 26. How many times is five go into 26? We know that is five times five times five is 25. You should. That is quite easily 26 months. 25 is one. We bring down the fire and we get how many times is 5 15 3 And that is our answer. So we've done long division life. You also know this example? For example, how many times the 16 going to 333. It doesn't go to 300 times going to 33 twice, Two times 16 is 32. Bring down the one I mean, Sorry. 33 minus 32 is one. Bring down. The three is three. How many times? A 16 going to 13 0 times. And we know that's remained a 13. Okay, but now we actually in the decimal section. So we want to divide a decimal number. For example, 765.5 by a whole number five. So let's go as normal. The thing you're gonna learn here is that all you gotta do is keep your dismal in the exact same spot I'm to do that for, You know? So five. How many times is going to seven Once One times five is five seven minus five is to bring down the six. We get six. How many times? Five. Going to 26 5 times. Five times five is 25. I say 26 months. 25. That is euro one. Bring down your five over here. You should not do this. Sometimes it's 5 15 3 times. Three times five is 15. So we know that makes no difference. 15 months. 15 0 We bring down five and Haman does times this 5 to 5 once. So that's 153.1. So now what? I want you to know we've done long division. Are we saying we have a decimal number? We dividing it by a whole number. All you've got to do is remember that the decimal keeps in line. Let's look at another example. 333.75 divided by 16. So how many times Study? Three. Going to 16. We know that's twice we've done those two times 16 years. 32 We say 33 months. 32. That's one would bring down our three. That's 13 Remain we keeping out decimal the same. How many times? A 16 going to 13 0 times, zero times 16 is zero. Now we say 130. We bring down a zero. I mean 13 with Bernama. Zero to make it 130. How many times the 16 go to 130? We know that 16 multiplied by 10 is equal to 160. So is lesson that what about eight times 16 multiply by AIDS. We know that's 80 10 times 18 80 10 and six times ages 48. So 80 plus 48 is 128. So we know that that's gonna work. And then 80 times 16 is 128. 1 30 minus 1 28 is, too. We bring down our seven. How many times? A 16 going 27 once. One time 16 is 16 27 minus 16. We know it quite easy that that is equal to 11. We bring down the five, we get 115. So we knew that 116 times eight. We knew that he was equal to 128. So it's gonna be 16 lesson that. So we can see that seven seven times 16 is seven times 10 is 76 times seven is 42. That's 112. So that's 112 year and remain. We got our decimal with minus stand. We get three. How many times that 16 Going to 30 now. We could keep doing this forever, but we're not going to do that. We're going to say this is 24 decimals and that's okay. But this, as you could see, could go on forever or a long time anywhere, Not necessarily forever. And you could solve this equation. But now how are we doing a question? A question like this where we said 76 point on 765.5 divided by 0.5. So this is no longer a whole number. Okay? We're gonna do that in the next video, so? Well, then, we've done dividing by whole number. I want to go back and divide by decimal now, because we in the decimal Fraction section. 41. Decimal Fractions (8 of 8) Finding square and cube roots: Okay, Great AIDS. Now we have the last video in the dismal direction section and that a square roots and cube roots, This section is fairly easy, but there's one or two tricks, so we're gonna go through carefully, right? You already know square and cube roots, for example. You know, the square root of 25 is equal to five, right? You also know that tweet. And you also know that five squared is equal to 25 right? You know that the Cuban of 27 is equal to three on. You know that three cute is equal to 27. Great. So what about decimals? How we're gonna use dismal. So what about 0.49? But we know that there's a number X squared, which equals 0.49 Remember that five squared is equal to 25. And here we knew that lots of five. But now we're looking for number X. So five squared is equal to 25. So if we say five squared is equal to 25 we know that five is equal to the square root of 25. When you move this across to that side can square to the other side goes to the square root . So what if we said that five X is equal to pass a minus the square root of 0.49? If you don't understand, the person minus you will understand in further grades. But bear with me now and you're sort of understand what's going on here. But we know it's the square root off 0.49 So we say, What is that familiar? That 0.49 is very similar. Seven times seven is 49? So seven square Those two sevens is equal to 49 so seven is the square root of 49. So what about 0.7 squared? We can quite easily see that 0.7 square to 0.7 times 0.7, and that is equal to 0.49 And that is what we're looking for. We can also say the negative, which I'm to show you now, but you don't need to worry about it. If we said negative 0.7, that's the same as negative 0.7 times negative 0.7. It's important to know that a negative times a negative is equal to a positive, which is exactly the same. And that is why we do the plus or minus over here. So we can say that X is equal to 0.7, which is equal to pass a minus 0.7. So that's the open seven or negative 0.7. So the square root of 49 of 0.490 point seven that works quite well. You'll also note, when we looked at ordering decimals, that 0.7 is bigger than 0.49 But here five is smaller than 25. So this is where decimals become a little bit tricky, but we're gonna work on it now. So that's why the answer would not be 0.70 point zero is him, and it's smaller than 01 49. Okay, so the trick for this is to always look for normal square and cube breeze, for example, 0.25 What if that we know that's 25. We know 25. Discredit 25 is five. So what if we tried 0.5? It's just check that 0.5 multiplied by 0.5. You can figure this out on a calculator, but let's do it like this. So we go 0.50 point five. We noticed two decimals there we multiply them five times five is 25. So we at the five we add the 25 times 00 plus two is two. Sierra is magic. 00 times 500 times zero is zero. So we know that that is 25. We have two decimal places. If you're not sure why we do this, make sure you watch multiplying of decimals. So this goes 12 and that's Europe went to five and we can add a zero there and that is our answer. So great. 0.5 works. What about the cube root of 0.275 and again you'll find that this is equal to 0.3 and you must taste that. Make sure you go 0.3 times 0.3 times 0.3 to test that answer. What about 1.44? Do you know your multiplication times table? What is 12 multiplied by 12 If we asked 12 multiplied by 12. That is equal to 144. So what about 0.12 multiplied by 0.12? Okay, so we said two times two is four. Two times one is to see we add image or that zero. We Adam is a two day We had a 01 times twos to one times one is one got to magic zeros. And then everything is multiplied by zero. So that's gonna equal Full two plus two is full. One plus one is one. Okay, now this is going four places across, So 1 to 1 to what's happened here? 123 There's We went there. 123 full. So wait away from here. A 1234 That doesn't look right. Zero quid. 144 What if we tried? 1.2 multiplied by 1.2. Let's try that. Two times two is 42 times. One is two. We had a magic zero one times two is two, one times one is one. It's Addie's all up for four one. We got two decimals across one to ah, that looks about rights. So is 1.2. Be careful with your decimals. Remember, it's only this year is smaller than that that is bigger on this year. Is smaller than that because it is less than one. Don't worry too much about that. What about 0.125 The cube root of 0.1 to 5 125? What does that remind you of? Five. Cute is equal to 125 so let's try 0.5 times 0.5 times 0.5. So what is Europe when five multiplied by 0.5? We've done that a radiant we know that that is equal to zero point 25 So what is 0.25 multiplied by 0.5? Because we times get by 0.5 times three. Remember it 0.5 multiplied by 0.5 multiplied by 0.5. What is that equal? So we first multiplied those to which of these two year to get that and we put that Then we multiply the pain. By the 3rd 1 Five times five is 25 to five times two is 10 plus. Another two is 12 we'll add a magic zero. We got 000000 You know that that is equal to 125. We got three decimals across 123 And that is where the equal 0.1 to 5. So it must equal 0.5. Remember, you always got to check in with decimals. You conscious guests? What about 0.9881? As you'll figure out that is equal to 0.9. And you can work that are quite easily. What about the cube root? Oh, sorry. But now these two I want you to look at What if they asked for 8.1 the squared of 8.1 or the cube root of 0.64? Thes don't work now. I'm not gonna get too much into the detail. Why? These don't work. But if you put this on the calculator, you're find it's a really big number that doesn't make a lot of sense at all. So it's important. You're just because you see a 10 or 12 Not just think it's going to be 0.9. And if you see a six and a four. Don't just assume it's gonna be four times four times for Be careful with these. Make sure you always show you're working out to check that your answers. Right. Okay. Well done. Great dates. I was squared and cubed. Roots. Now we're gonna go into rate and proportions before last the financial mess. 42. Rate & Proportion (1 of 4) Comparing quantities of the same kind (Ratio): Okay, great. It's now we are doing a section off rate and proportion, and it's mostly doing with ratios rates, sharing ratios and or increasing and decreasing ratios. Proportion is a fancy word for ratio. So these are the exact same by the in here. We're gonna fall in some words yet. Okay, what is the ratio? Racer is a comparison of two or more quantities of the same type. All measurements, right. It's a proportion or ratio, but they must be the same time. If they're not the same type, we're gonna have to change them to make them the same time. So we write the numbers in a ratio with a colon between them. For example, if there are six girls and eight boys in the bus, we see say that the ratio off. Six. The ratio's off six goals to every eight boys in the bus, and we have right that as 6 to 8 aeration six goals to eight boys. That is another way to do it. Six girls to eight boys. They're both Children or both people, so they're the same type right type, and if you notice with 6 to 8, it's a similar or exactly the same as saying 6/8 for every six goals EG Boys rides. And how can we simplify this? We're not gonna go into detail with that. No, but that's the same as writing 3/4. The ratio is exactly the same as three goals to every four boys. Even though there are the total six schools but girls and eight boys, the ratio is exactly the same, so we can simplify a ratio. Ratios don't have vision units because the units cancel out. For example, if we had a ratio off two liters, 23 liters, we can say that we have a ratio off 2 to 3 without writing the leaders. The units must be the same, however. For example, if we had a ratio of 200 millilitres milliliters, you know familiarly to is a can of Coke is about 330 milliliters. So 200 milliliters 23 liters. We can say that we have a ratio off 200 milliliters. 2 3000 millimeters. Why? Because there's 1000 milliliters in one meter rights, So three liters is exactly the same as 3000 milliliters. Okay, now we can simplify by cancelling and Norton either side 11 When they weaken, do another one, and that is the same as 2 to 30. We can simplify this by writing. It's even further in its simplest form because if you divide both sides by two, so two divided by two is equal to one and 30. Divided by two is equal to 15 so we can write it like that, and that now is caught its unit. So let's write some examples of ratios in the simplest form. What about five and six? That is the same as divide both sides by five. It's one. What is six divided by 5 12? That is its simplest form. What about three and 30 again? You can see that we could divide both sides by three to give us one and 10. What about 16 and 18? Ah, let's start by dividing both sides by 2 16 Divided back to is eight. An 18 divided. About two is nine. Ah, that is the simplest form we can get. We're not going into units. Yes, but this is 8 to 9. We'll see units on the next slide. What about 680 weaken. Divide 180 by 66 Divided by six is one. What is 180 divided by six. We know that six times 10 is 60 times 2120 times. 30 is 180. Pause the video and try these by yourself for this last 1 44 and 55. Can you figure this one out? Can you find a lowest common denominator between these two? What about 11 44? Divided by 11 is full and 55 divided by 11 is five, and that is the simplest form. So what about a unit? Duration. Sometimes you write a ratio in unit form. This is where the one side is equal to one. Remember unit means one. For example, four and 16. We could write as one two full, and that is a unit racial. Sometimes it won't be in its simplest form, but we still need it in a unit. For example, fall to nine. This is the same as one is to 2.25 How did we do that? Well, remember here to get this to foreign. 60 and we said four divided by four. That's it one, and that's how we got the one. And then we said 16 divided by four and that's equal to four. And that's how we got the four here. So we're gonna do the same principle. We divide both signs. So he said, four and known is the same as four. Divided by four and nine divided by four. Because we divided by full, we divided by full and that is equal to one and 2.25 You can quite easily see that nine over full if we have to write it like those 9/4 is the same as to and 1/4. And what is 1/4 we know that is 0.25 on That is the same as two point to five, and that's our answer. One is to 2.25 See, now we got ratios with decimals. Yeah, okay, so let's write these in the unit form. What about four and 60? That's quite easy. We just divide both sides by 44 Divided by four is one of you had to get it to one 60 divided by Force 15. That's easy. What about 13 and 39 is divide both sides by 13. That's 1 39 divided by 13. That's three. Okay, there are 32 teachers at your school and 992 students. What is the unit ratio we see Unit. One of them have to be one off teachers to students. So it's a 300 city on 32 to 9. This 32 teachers to 992 students, 32 divided by 32 992 divided by 32. And that is equal to one and 51. For example. What if he said, get three to fall into unit ratio? So we divide the left inside three by three, and that's very easy. Equal to one and four by three. The right inside or what is four by three? So we know that four by three. So it remained. We said 3/3. That's equal the one that's easy. What about We go for over three. So we said forever. Three is the same as one and 1/3. What is 1/3? We know that 1/3 is there a 10.33333 recurring. It's a terrible three. So we can say that is one is to 1.3333 and can. So this is how we do ratios. Remember, Russia is a comparison of two or more quantities off the same type or measurement, and it's a comparison center ratio. It's a proportion, and that is our ratio. Now we're gonna move on to rate. 43. Rate & Proportion (2 of 4) Comparing quantities of different kinds (Rate): Okay, Great dates. Now we're gonna talk about the rate remained reading ratio and the ratio was a comparison off. Two more quantities of the same type. Now we're gonna fall something in your when we done with regards to rate. So great is the compassion of two more numbers or quantities. But the two numbers have different units. They different then no longer the same, right? So, for example, you might have seen those very, very common form of a rates. Sarkar travels 60 kilometers in one hour. We can say that the car travels 60 kilometers per hour. You might see a speed limit that is no faster and 60 kilometers per hour. So that means in one hour the car would travel 60 K's. How far will travel in two hours. That's quite easy. If it's traveling 60 kilometers every hour and is doing two of them ever, then travel. 120 cares. Okay, so that is our rate. Kilometers per hour. Okay, lets say now a car travels 80 kilometers in two hours. We can say that the car travels 40 kilometers per hour because in each of those hours it probably average 80 air 40 K's, even if the one hour it was 41 the other was 39 K's. An hour care bag of carrots cost 22 rand per kilogram. You might have seen this in the shopping center that is the same as 22 rand per kilogram. Okay, so how much would two kilograms cost? So if we knew that one kilogram cost 22 rounds, what is two kilograms? And that is easy. You can probably work out that that is 44 round. Okay, How about how much would 0.5 kilograms costs? You again could see that if one kilogram cost 22 round, half the kilograms would cost half the price. So that is 11 runs. But how would we do it in more detail with equations? What if we ask for 0.45 off a kilogram or 0.444 kilograms will deal with that? In the next example, an athlete runs 100 meters in 20 seconds. What is this Speed in meters per second runs 100 meters in 20 seconds. So how many meters per second does your own? So he said he runs 100 meters for 20 seconds. 100 meters for 20 seconds. So what is his meters per second? Right to get it two meters per second. We need a whole number here or not Mystery A whole number. But we needed to be over one at the bottom here. So to get this over when we go divide by 20. So we say 100. I have a 20 divide by 20 to get the bottom 21 and that is equal something of a one. And then we divide by 20 in the top. What's 100 divided by 29 is five. So he's going at five meters for seconds. What is this? Speeded meters per minute. So if he runs 100 meters in 20 seconds, what is the speed in meters per minutes. So we going 100 meters right in 20 seconds. How many seconds is the other? In a minute? You know there's 60. So how far will you go? In a minute You're times by three times by three and that is equal to 60 seconds over 100 times three is 300 meters in 60 seconds, which we know is the same as 300 meters per one minutes. So you can say that he is going a 300 meters per minute. So this is the important part of a rate question. How long would it take to do one kilometer at the speed? Okay, so we know that we know how long he have fast is running. We know he's running a five meters per second. So how many seconds will it take to run 100,000 meters? Because we know that one kilometer one kilometer is equal to 1000 meters, right? And we know that he's going at five meters per second. So what is 1000 divided by five? You know that 100 divided about five is 20 so that must be 200. So 100 meters per second. We divide that and that is 200 um, seconds. So he's gonna take 200 seconds. Remember that that is meters her second so in the region, this he's going all right. Let me just rewrite this because that is meters per second. So this is 1000 meters over size meters per second. The meat meters cancel each other hours. That gave us 200 and divide by divide is equal to a positive. Oh ah, multiplication. It's the same. And that is or sorry. A one or about one over is the same as that number, and that is 200 seconds. Don't stress too much about that unit, so he's gonna take 200 seconds and what's at the same. That's three minutes and 20 seconds. Okay, let's look at this example to better Understand this. A golf cart drives 3.2 kilometers in 10 minutes on a run arounds 200 meters and 40 seconds . Who is faster? So we're using Qalat, kilometers and meters and militant seconds. What do we know about rate? We have to get them to the same unit. Type on a type of the same, you know? Yes, unit type. It must be the same unit time. So we say 303.2 kilometers to 10 minutes. We divide each side by 3.2. So 3.2 kilometers divided by 10/3 0.2. So 3.2 divided by three countries. One 10 times treatment to just still in minutes. Do not forget that rice and there were gonna go. We might have to drive long division here. So you say, What is 33.2 divided by 10 remain. We're gonna move the decimal one across to, say 32. Provide by 100. Remember this if you're not sure, make sure What's this video? How many times a story to go to 103 times Three times 32 is 96 1900 minus 96 is full. We haven't decimal here. Remember this most? Here we can add A whole bunch is euros if we want to. We bring a zero down to make that 40. How many times is 32 going to 40 once, one times 32 32 40 minutes. 32 Aides. We bring another zero down to make that a 0 32 goes into eight twice. Two times 30 to 64 18 months. 64 is 16 within the last year. Oh, down to make that is your own 3200 times going to 160. 32 goes into 165 times. It's just check five times 2 to 5 times 30 is 155 times to his 10 150 plus 10 is 160 brand . So that is our answer. One kilometer to 33.125 minutes. Okay, Now remain. That's not 3.12 seconds as 3.125 minutes. Okay, so let's try the other side now, 200 meters to 40 seconds in 40 seconds. So that 0.2 kilometers in 40 seconds. Because we know that a meter there's 1000 meters, like I milliliters example in one kilometer. So we just convert that two kilometers so we can start off there. We got that. Great. Now we need to get this side. And when you get that tow one, how do you get this toe? One. We go 0.2 divided by 0.2. That's easy. That gives us a one. What is 40 divided by 0.2 40. Divided by zero point Again, we're gonna have to go, uh, 0.2 into 14. Let's move that to across to into four hundreds. Well, that should be quite easy. Tune to 40 goes 20 times 2400 goes 200 times, says one planter in 102 100 seconds. We need in our convert 200 seconds into minutes only. I guess we know 200 seconds is the same as three minutes and 20 seconds. We need to get that into decimals. We can't have the second year. We only one minutes. We know that 20 divided by 60 seconds cause it's 60 seconds in a minute is 0.333 So we can say that that's one kilometer in 3.33 minutes. So which is faster if you did? One kilometer in three minutes, 30.1 to 5 or one counter in 3.3 minutes. This is clearly the smaller number, so it is much faster. So the golf cart drives faster. I guess we did rate ratio in limbo. Now the race. The rate is a comparison between two numbers, but the numbers can have different or do have different units. Now we're gonna move into sharing and racial well done. Great. It's 44. Rate & Proportion (3 of 4) Sharing a giving ratio: Okay, Great. It's now we're gonna talk about sharing and ratio, right? So, race, you can have more than two quantities. What I mean by that? Let's have a look at some examples. Example one. A bag of fruit has 20 apples, 15 oranges and five bananas. We can say that the ratio of apples two oranges to bananas is 22 15 to 5. This can be further simplified by dividing by five. Right, so we can say 20 divided by 5 15 Vital but 55 divided by five. And that is this 4 to 3 to one. So, for example, if I had four banana and four apples, how many bananas would have? So we know that those are apples. We put a day, those are injured and those been honors. So we have four apples. So that's easy. That's a really full How many bananas would have just one? So then we would have won if I had 12 bananas. How many oranges would I have? Okay, so now bananas needs to go to 12. To get bananas to 12 we have to multiply everything by 12. So oranges by 12. So then we can get four times 12. What is four times 12 40 plus and eight? That's 48th. So 48. What is three times 12? Three times 12 is 36 what is one times 12 is 12. So we had 12 Wynonna's. How many oranges would you have are 36? We didn't actually need a workout that 48 but I've done it for you anywhere. 36. Okay, so what are these? Ratios are equal 4 to 60 and 2 to 30. We can quite easily see that if we divide each of these by two, we can have 22 30. So those are equal. What about 3 to 60 and 2 to 20? Is it possible to get that? The same is that let's simplify. Both of these much as possible is divided by 300 side of year. So divide by three, we get one to 60. Divided by three is 20 and it's divide by two year, and that is 1 to 10. We can see that those are not equal. Okay, what about one is to 40 and 1.5 is to 60. Let's get us to 1.52 Unit How do we get that to unit? We divide by 1.5 on both sides. 1.5 divided by 1.5 is one what is 60 divided by 1.5. If you're battling, you might want to go 1.5 into or 60 in 211.5 to 16. We know that's the same as 15. Into 600 15 into 60 is four times four times 15 is 60. We get zero, We get 0 15 into zero. Go zero times 40. Remember decimals. They know. So this 60 divided by 15 is 40 I Those are exactly the same. So this is exactly the same. Yes. Those are what? About 18 to 30 and 5 to 7. Okay, let's simplify. That is a simple as we can get it for No, Mr Vaidi, side by three. So divide 18 by three. We're gonna get six. Divide 30 by three. We're gonna get 10. Okay. Can I really see that? Don't look like they're going to be the same as Divide that by two and get three and fines . Sen. We can't see that those the same. But let's try and get it into a unit fraction anywhere. So that's three and five. So remember, we work with this 18 to get to that, it's divided side by three. So that's gonna be three divided by three 25 divided by three. And that is gonna be one is equal to 5/3. And here we divide the side by five. So five of five to save in over five and that is one is to 7/5. And we can quite easily see that these are not the same. If you had put that five divided by three and a calculator USC that those are different values they're giving you. It's quite simple to figure this out because you could say that is the same as one and 2/3 and that is the same as one and 2/5. Clearly our difference on the quickly clear the screen. Okay, let's look at another example. They are 2 to 3 females, two males in a class and there are 15 males. How many females are there? Okay, so for every two females, we have three males and we are told that they are 15 males. How many females are there, so two females to every three mills. So that's up 2 to 3 ratio. So we know that we have 15 males. How many females will be here? So what do we do to get from there to there? We said three multiplied by five. So then what do we need to do to get from there to there? We need to say to multiplied by five. That's what we do. And we get out on Soft 10 to 15. So when there's 15 males in the class, we know that there are 10 females, so they're 10 females in the class. Let's have a look at one last example. A bag of cement is four kilograms of sand, 2.5 kilograms of play, 500 grams of stone and one kilogram of line. What is the total weight of the two bags short? So the total weight of the two bags so we can tear this ratio, for example, might be different. But let's figure this out. Four kilograms, right? Four K, Jeez, plus 2.5 k Jeez, Plus what is 500 grounds? You know that there's 1000 grounds is equal to one. Sorry, one kilogram says 500 ground, 0.5 kilograms fast. One kilogram. So four plus 2.5 to 7.5 plus 0.58 plus one is nine. So there's nine kilograms in total for one bag per bag. So we're gonna say easy. Multiply that by two to get 18 kilograms for two bags can I was quite easy. What is the ratio? Right? So we had to put this in 0.5 to remember. Ratio is the same unit. Kilogram, kilograms, kilograms, kilograms. That's perfect. So the ratio off sand to play two stones to line Is that this four year? What if we wanted to make the unit ratio for one kilogram of stone? So we wanted those stones over here. It to be one. How do we get 0.5 to 1 going to get that point 0.5 times to get us one? So we gotta multiply everything here by two, and that's gonna be eight. And that's going to be 2.5 times twos. Five 0.5 times two is one and one times two is, too. And that is that awesome. And that is sharing a ratio well done so racial can have more than two quantities. Don't forget that, for example, years three now last section, and that's increasing or decreasing ratio. 45. Rate & Proportion (4 of 4) Increasing or decreasing a given ratio: Okay, great. It's now we're gonna talk about increasing or decreasing a racial. And what does that mean? Right, So ratios can change when you add or take individual units. That sounds a bit random. A bit of a weird thing to say, but we're gonna get into it and explain exactly what the sentence means. It probably makes sense already, though, Right? Let's go through some examples in your cross country team there. Seven boys and 10 goals. Really, What is the ratio of boys to goes? So we know that that is seven to arcs. Chemical on 10. They're seven boys to 10 goals. Crimes. What is the ratio of girls to boys? That is quite simple. There 10 goals 27 boys, Just to show you that we can switch a rash around. So it's important which order the ratio is in now. If three more boys joined the team, what is the ratio now? Right, So we add three boys, so there's gonna be 10 boys right and 10 goals. So the ratio of boys to girls is 10 to 10. That's quite easy. We can simplify that by dividing 10 on each side, and we get one 21 and will be both the same way either side. So one boy to one goal, one goal to one boy. It's the same here, Right? Let's have a look at another example. You have a bag of 50 strawberries, 12 or Rustem. What is the ratio of good strawberries to Russian strawberries? So we know that they are 50 good strawberries or 50 strawberries, right? And 12 a rotten. So take away 12. See, we take away 12 and we're gonna get 38. So the ratio of good strawberries too rough and strawberries are 38 good strawberries for every 11 right for every 12 Russian strawberries. Okay, now, what if I remove that a Trafton, strawberries or eight of the rotten strawberries? Rather doesn't mean the eight and 10 good strawberries watches them Eurasia. So these are good stories in these are rotten strawberries. We say 38 we take away eight good strawberries, and that is 30 and we take from the 12 Russian stories. We are sorry we take away 10 year to get us 20 aids who took away 10 good stories and from the 12 Ruffin stories and take away age to give us four Russian strawberries, so the new ratio is 28 to 4 Ryan's It's worth now, looking at these are different ratios. What is 38 12? Can we simplify that? Let's start by Harding it to get us. What's 38 Divided by two. That's 15 is the half of 30 there. Half it is 4 15 1st Force 19 26 Okay, And what's half of this least half that. We get 14 on the side and two on the side and can have it again and 7 to 1, and we can see that the ratio of 71 is very different to 19 to 6. So here we have decreased the ratio of Russian strawberries. So we might decide if we were producing strawberries that if we could get rid of rotten, ate rotten strawberries and by mistake, have machine text. Wait 10 good strawberries. We've not decreased the racial rotten stories, although we do have fewer strawberries. Okay, let's look at another example. I have two bags of fruit, one with 21 pieces and one with 35. The ratio of apples to oranges on the first bags, 2 to 5 and 5 to 2 in the second bag. If I attitude that's together, what is the total ratio? Will the total ratio be 2 to 552 or 2 to 2121525 What will it be? We conscious added together, we've got it, actually separate each of these bags. Okay, So in our first bag, we had 21 pieces of fruit for every two apples. They were five oranges. Ryan's. So now we need to figure out how many in total we had. So we know that we at least at two apples, the cycles and the sun oranges, we know that two apples and five oranges. So this is a total of seven. Then we add another two apples and five oranges. Okay, We had in our total off 14. That. Plus that's okay. Let's have another two apples and five oranges. And now we get a total off 21. Ah, that is off 21 pieces. Okay, so we know that in total, they'll six apples in the 1st 1 and 15 oranges, and that you could see equals 21. Okay, so we have six apples and 15 oranges. Let's look at Bagram to I'll second back. We have 35 um, apples and oranges in total. We know the ratio is 5 to 2. So we have apples and oranges. We know that for five apples we have two oranges again that equal seven as convenience. We have five apples again and two oranges. That is seven again. So that the total is that first that is equal to 14 five and two. Again, we could see that that is another seven plus the 14 that is gonna equal 21. Okay, another five to another five and two up seat to other five and two. Is that a seven seven? Well, that is 14 plus 21. And that is our 35 that works well so we can see here. We have 25 apples and we have two plus two and 10 oranges. So in total, if I added the two birds together, I would have six first fire. That is 31 apples for every 15 1st in 25 oranges. And that is our ratio. No, not that we can't simplify that any further. So Russia has changed. When we increased or added two bags together. Well done. Great. So we've seen now that ratios can change if you add or take away individual units. 46. Rate & Proportion (1 of 4) Comparing quantities of the same kind (Ratio): Okay, Great. It's now in quite a fan section or interesting at least court financial mess in the sexually gonna be dealing with profit and loss discount that simple interest, hire purchase loans and exchange rates. By the end of this video, you're gonna have fold in a formula right here in the profit and loss section. So let's start when we buy something, we buy it for a cost price rights. For example, If I buy a pain for nine rand or if about apple for $1 that is the cost price of that object. When we sell something, we sell it for selling price. Okay, so if I sell a pin for nine around or for subtle an apple for $2 that is the selling price and then we buy something for the cost price. So that's what we use. If selling price is greater than cost price, that is a greater than Remember that the biggest side day is the bigger object. Here we are making a profit. Okay, If my selling price is less Emma cause prize i e. If selling price is smaller than the cost price, which is bigger, then we're making a loss. Okay? And that's a profit and loss section. So gain is the difference between a selling price and cost price. Saw a gain. Also notice profit. We're going to speak. Gain and profit is the same thing. Is are selling price minus our cost prize. That made sense, Right? Let's have a look at the percentage here it is the percent off gain over the cost price. Right. So the percent gain all percent profit. They're the same thing. Is the gain or profit divided by the cost price? So this close to be a percent loss if we're making a loss, so I eat. This would be when the cost price is greater than the selling price. So the last would be the cost price minus the selling price only when the cost price is bigger than the selling price. So the percent loss would be the loss over the cost prize. Let's have a look at some examples. What profit or gain done make of our by a pack of chips for four around and sell it for five rooms. Okay, so we first want to know what is the prophet of game and then what is the percent profits. So what? What is gained or profit? We know that game all profit is selling price minus costs prize. And we know that that's gonna be the selling price of five round minus two, selling right part of the cost price of four own. So we're gonna have a gain off one ran here. That makes sense. If we buy for four end on, we sell it for five grand. We have made a profit of one rude. So what is that percent profit? That person profit is the game over the cost price. What is our gang here again? Is one round one round divided by four round? Has he killed the one ever full and that is equal to 0.25 If you put in a calculator or I'm sure you can see that is 25% south of ST profit is 25%. So what Loss die make if I buy a pack of chips for five grand and sell it for three rounds . So this is not great at all. We've both the chips off Iran, and now we're gonna sell win for three road. That's not very good business. So we're gonna make a percent loss, right? So that mass now is the cost price minus the selling price. So the cost price he knows five rand on the selling price we know is three rooms. So we have lost to Rand in this endeavor, right? So that percent loss is the loss over the cost price. So what is that to rand divided by five rounds. And this is going to give us a loss off to over five, which is the same. A zero point fall, which is the same as 40%. Let's have a look at another example. If I buy tin hamburger Patties for 25 runs and 10 hamburger rolls for 15 rooms, what must have charge for each hamburger to make a 25% profit? So now we not told what we're gonna sell each of these hamburgers, but fall. But we rather told what are percent profit must be. You can tell that 2% and it's the percent profit. So let's have a look. What is the cost? First of each hamburger. So we know that a hamburger is a patty casserole. We have 10 hamburgers, Patties for 25 round and 10 hamburger rolls for 15 road. So the cost of each hamburgers the cost of a patty has the customer role. So that will be whatever the cost of one patty be, it would be 25 divided by 10. Because we know this 10 hamburger rolls cost is 25 rounds, and the cost of the roles will be 15 ran, divided by Tim and that is equal to two run 50 plus one round 15. So each hamburger role is gonna cost us four round to May. Okay, so we want to make a 25% profit on this four round South person profits is equal to gain over our cost price so we can change this formula around. What I've done here is I've moved the cost price two year. That is exactly the same as times in both sides by cost price over one and cost price over one. This is basic function manipulation that I'm sure you do know otherwise your videos to help you with this So the prophet is equal to the cost of the percent profit times. The cost price is equal to the game. That's perfect cause. We need to figure out what the game is because we know the cost price and we know the percent profits. So the game is equal to So I've just moved these around here, so the percent profit is 25% and the cost price is four end. Okay, so I'll gain is one runs. So then we can quite easy work out whether selling prices, because we know that gain or profit is equal to selling price minus cost drives so we can change it around again by moving the cost price Over here on, we say the selling price is equal to the cost price plus the gain or profit, and that's quite easy to do. So we say the selling price is four round plus one road on. We know that that is equal to five rounds. Okay, so that is our selling price of each hamburger patty to make a 25% profit. Now we've learned that the person profit is gained over the cost prize and the perceived loss is the loss over cost price. It's always over the cost price. Okay, Well done. We got a non. We want to discount very on discount in a video. Invest 47. Rate & Proportion (2 of 4) Comparing quantities of different kinds (Rate): Okay, Mathematicians now we're gonna be doing discount and that. And what is this fact term that you might have seen on shop shelves? So let's start with discount and let's take 10 rounds. What is 50% off 10 rows? When I said What is 50%? It's the same saying What is 1/2 off? 10 runs and 50% 10 round turns, 50% is equal to 10 ran time 0.5 and that is quite easy to see that that is equal to five around what is 10% off 10 rooms. Again, it's the same saying What is 1/10 off? 10 round? And you can probably work up that 10 grand times 10 percents is equal to 10 ran time 0.1, which is equal to one runs what about 14% of 10 run? That's 10 rand turns 14% which is equal to 10 round by 0.14 Remember that a percent is always shown up like that, with two decimals to the left because of the same deserve 100 which is two decimal places to the left. So 10 random 0.14 and that is equal to one round 40. Do you get the idea what we're doing here? We're taking what is 50% of 10 and we can say that amount of money times at present on. Then we simply convert that percent into a decimal fraction. What if I said what is 150% off 10 rounds? Let's do the same thing. 10 run times, 150%. And now this is equal to 150. We move it one, we move it 12 and that's 1.5 10 times. 1.5 is 15 runs. That works just as well. So what if I want to ask you what is 14% of 12 around 50. Could you work this out? We go tolerant 50 times 14%. That's the same as 12 and 50 times a record one fall. But what if you looked into your bucket and packet of things? You knew what to do, and we could do long multiplication year. If you haven't watched the video on how to do long multiplication with decimals, make sure you do. I'm quickly show you here. You can always do these questions even if you don't have a calculator. What is four times five? We know that is 20. Remember long modification. We stack everything along the side. We don't have to line the decimals up. Four times five is 20. We put zero. We added to four times two is eight plus two is another Tim. We're gonna tuck, put that zero there and add a 14 times. One is four plus. A one is five. We add our magic zero. What is one times five? That is five. What is one times two? That is to what is one times one? That is one. Now, we could add up to magic zeros if we wanted. And then zero times, nor 5005 to 00 times one is zero. So that will give us all zeros, Which doesn't really matter. So let's add these up. We have zero. We had this up. We have five. We add this up, we have seven. We had this up. We have won. And now we can't. The decibels we go. One, 23 These three decimal. So go. 123 and then we can tell that that is equal to one round 75 and that is 14% of 12 and 15. So I want to give my friends 25% discount on, like, 12 round pins, so I want to give them a 25% less off 12 round for my pens. How much do I charge them? So what is 25% discount October end? That is 25% of 12 rounds, which is 12 times 25 which is 12. Son, 0.25 which you can probably quite easily see that that's 1/4 of 12 and that is equal to three rounds. But remember, we're not charging our friends. Three round were rather charging them. Three ran less and 12 rounds, so we can simply go 12 minus three to equal. I'm nine rounds, but there's an easier way to do this. What if I said 100% minus might just come price? So I said 100% minus 25 cents, and that is equal to 75%. And then I said, what is 75% off 12 runs and that is equal to 12 times 75% and that is equal to 12 times 0.75 which is equal to nine round, and that's a much easier way to do it. Although they go, both will give you the same answer. Let's say I want to give my staff a 16% discount on my 67 round hamburger and chips. What I charge them. Let's go 100% minus have discount percent, So that's 100 minus 16% discount, and that is equal to 84%. So what is 84% of 67 round we can go 84 67 round time, 0.84 And remember, we can use it like this if we don't have a calculator with us. Our answer is simply gonna equal 56 around 28. Don't don't forget, you can use this long multiplication method Right now. This look invests that is value added tax. That's what the V 80 cents for its value added tax. And what does this mean? That is a form of tax we pay for when we buy goods or services. So every time you buy anything at the shop, unless it's milk or bread, you're paying value added tax on it now it is different in every country. Your vets. It is different in England, it is different in America. It's different in every country in the world. In South Africa, that is 14% right. So that means a shop owner will add 14% that onto all their products. On this, 14% will be paid to the government. It is a tax. All right, Don't forget, it is a text. Let's have a look. At example, you want to sell your pants for 10 rounds. 14% of this will be that. What fact? You pay if you're selling the pains for 10 years. So this is money that you will be paying the government. So 14% of 10 round is 10 grand times 14% which is 10 random zero point fall, which is one round 14. So you gotta take this into account from your trying to calculate your profits by remembering that one rand 40 of all the pens you soul will be going to tax for the government A callous, continuous and dissolved Well, let's say you bought a packet of chips for three ran and you want to make a tour and profit after. That's how much should you charge? Why? Ideally, you would want to sell the chips for five grand. That's obviously three Ranjha board than four. That's the tour and profit, and you think, Consultant for Fire and then you make a to ram profit. But that is not actually true. That is not the case. You'll pay a vat on this, so we need to add 14%. And it will always be 14% when you're dealing with problems regarding South African goods and services being sold in South Africa. So let's add that 14 sent. You want to make your customers pay that 14% so it doesn't dig into your profits. So that is five grand times, 14%. That is five grand time 0.14 which is equal to 0.7 rand's or 70 cents. So the selling price, including that is equal to the selling price. Before you add that pass, your vet does that make sense. So what would that be? Your selling price, including that is your firebrand, which is selling price before. That's sorry, that's your 70% vet. Sorry, I just meant to be a zero here. Plus you're 70 cents that sorry that is meant to be a zero. So you're selling price, including that's is set five around 70. Let's have a look at another example. What that do you pay on a 200 round cricket bets? And now you're buying of cricket bets? How much fat are you paying to the government when you buy the cricket bats? The fattest turned around times 14% So that's 28 ran. You know how to work that I have? No. So sometimes the price will say, excluding that. It's very rare to see that in this country, but sometimes you might see it. So what if you says there was a soccer ball that was selling 450 relative excluding vets, so you would go and want to buy the soccer ball? You would actually paying the shop owner more than 150 rand because they were so add VAT onto this price. So how much will you pay? What's 115 grand plus times 14% 150 times? Airplane 14 is equal to 21 round. You gotta not add the 21 roads to the selling price. So the selling price, including VAT, is a selling price before, that's that we know the 150 rounds plus a vat of 21 rooms, and that is going to be 5150 plus 21 which equals 171 red. And that's how much you will actually be paying. So look out. If you ever confined something that says, excluding that, it is very rare. So now we can add the equation with that all now we can add that 14% in South Africa, always on the selling price, including that is equal to the selling price Before that past the vets. Well, then I'm gonna move on to simple interest. 48. Rate & Proportion (3 of 4) Sharing a giving ratio: Okay, great. Let's Now we're gonna be doing are simple interest we're gonna have full in a block here. And it's gonna be our first identity. We learning in financial mess. Okay, let's talk about simple interest. Let's take 1000 rounds and invested for one year at a 6% interest. So how much is going to be there? After one year, the interest is going to be 1000 rand times where 6% and that is 1000 times by six. Over 100. That's 1000 times by 0.6 You know how to get from a percent to a decimal and that you can see that the interest gained is 60 rounds. What do we do year? We simply said we call that interest, but it was simply 6% off 1000 round. We found to be 60 rounds. Okay, so the closing balance is how much we have at the end of the year for opening balances. How much we started with and the interest is how much we gained during the time period and in this case is one year SARC rising balance is equal to 1000 rand. That we started with We invest that we started with 1000 rounds plus our 60 ran interests that we made in the year and that is very easily gonna be, as you can see, equal to 1000 six and 60 round. And that is our closing balance. Okay, sometimes we're gonna call out opening balance. R P O r. Principal amount. This is how much we started with how interest rate we're gonna call a little baby I So this could be 6% per annum per annum means per year p a. That's per year, 12 months. We might take her month, which is p in. So we'll always have the value after the percent. It's very important for that or could be per day. In a very rare case. The interest amount is I capital I this the mountain interest you have gained. So, for example, interest waas when 1000 rand tons six. You can see the I they and that is I equals our principal value of 1000 times are 6%. As you can see the principal value in here the 6% When here and I was p times a gun. I was p times, baby I and are closing balance is equal to our opening balance. Pass our interest. So this is our closing balance, as we've learned is are open balance which was principal times are interest. Yeah, which is p times I rides, right, So this is the same. If I made another ST Pierre would be, say, Mazar opening balance are opening balance, which is up here classle interest, which I was Ah, I And that is the same as we've done here, which is equal to P. And what is I equal? P times I plus okay, multiplied by I We can simplify this fraction to make it equal p one plus I cause we know that that is the same as p times one. And we take out a common factor. So that is the same as p multiplied by one. Plus he times I if you're not sure what we're doing here, make sure you watch the relevant videos to practice. Take your common factors in functions and equations. And then we take out pizza common factor and that is equal to p one times multiplied by I. And that's exactly what we got here, right? So let's create the identity. I'm gonna tell you what it is. Our closing balances P weapons I So we're gonna say a is equal to P one, Plus I end So the only difference between are closing balance, which is that we added it in and we added a So we call you are closing balance A. You can. And then we added in Why are we doing this for A is equal to our accumulated amount, It's our final amount. This could also be called your closing balance, but we're gonna call it a P U knows your principal amount is your initial value I as your interest written as a decimal, very important that it's as a decimal and in is the number of years, you know, it will be the number of years because the interest will be per annum. Right? So let's have a look at example hazy Could a P one because I turned in You must remember this. This is your identity. If you invest 2500 inter bank accounts have 4.5% per annum, simple interest rates How much will you have the end of three years we have a three years we can sell to add that so we can say is equal P. One past times in P is 25,000, 2500. That's our principal value. We added. Our one I. It has to be in decimal. We know. How do you get a percentage into decimal? You divided by 100? So we said 4.5 divided by 100. That's great. And then three years that is are in for number of years and that is equal to 25,000 person times. One has syrup went zero for five times three, which is one past 0.135 was equal to 2500 times 1.135 which is equal to 2837 around 50. And this is a This is not accumulated total or our closing balance. Let's have a look at another example equals p one has items in. Keep reminding yourself of this. This is gonna you're gonna remember this. I borrowed 300 ran for my father at a 6% per annum. Simple interest for six months. Mm. You're talking per annum. which is per year. But now we have months. We're gonna have to think about that. How much will at home at the end of six months? Okay, Hey, equals P one pence items in, and now we're gonna have to change our end to be 1/2. But let's go through. P is equal to 300 right? That's up. Principal value are starting value. We have our one. We haven't interest rate in the decimal by taking the percent in the violent by 100 now in is gonna be have to be 1/2 because only half a year we must not use six because it has to be in terms of number of years. So don't forget that. Ankles. 300 times 140.6 times 0.5. You know that 0.5 is the same as 1/2 as you were 301 Casino Quinn, 03 And that's stranded times 1.3 At the end of six months, we're gonna earn 309 rounds. Let's have a look at one last example as he could've p one plus items in keep remembering that? How many years must you invest? 4000. Read it at a simple at a simple interest rate off 7.5% per annum before close to 5500 rounds. So we're gonna start with 400 we know that that is up. He we know interest rate. I we know our A because that's our final A So we got a we got p Got I Are we gonna have to solve in how many years? Great. I can't equals p one plus items in. So we know that a is 5005 hundreds. That's what it's gonna end off at before it grows to that. We know principal value of 4000. You know, one we know interest is 7.5% divided by 100 to get it into decimal fractions. And there were times about in a number of years. Okay, so now I'm gonna manipulators. You know, you can take the 4000 down to become 5500 divided by 4000. You know this if you haven't, If you're not sure, how are we doing this? Make sure you watch those videos of manipulating equations and they will be left with one plus 0.7 That's this bracket here times in. So that's 1.375 That's this number one has. We've dropped the brackets here. We now take the one to the other side and make it a minus. It's keeping those exact same. So 1.375 minus one is 0.375 and that is equal to 0.75 times in. Now we can take this again to that side to make it zero points 375 divided by 0.75 And now we can sell for in in is that I've just such search around. They have done nothing to the equation. So n is equal to five years. Well done. We can now full enough. Some winters form with subway interests is a close balance, which was the opening balance plus interest. And we know our first identity and that is a accumulated total is equal to p. Ah. Principal value are starting value. One has IR interest times in the number of years 49. Rate & Proportion (4 of 4) Increasing or decreasing a given ratio: Good morning, everyone. Today we're gonna be doing hire purchase. We still in the financial math section, and by the end of this video, we're gonna fold in something. This block here we're first on profit and loss, discounting, vet and simple interest on. Now we're gonna talk about hire, purchase and what it is. So what is Hire, Purchase, Hire, purchase or we call it H P H dot P and capital. It's a nice way to donate. It is a way of buying expensive goods using high purchase, you pan initial deposits on the goods and then the remainder in monthly installments. Goods will normally cost more this way, right? So let's look at an example to better understand it. You want to buy a car for 20,000 rounds, but you only have 5000 rand saved up for the car. You can't buy the car, Yates. You're earning 4000 rand a month. So this tells you you don't have enough money to buy the car. The car company might allow you to buy the car on hire purchase. This means that say you need the car to get to work. You can still buy the car now, but they're gonna get their profit from it by charging you even more. Although you pay it in monthly installments over time, rise. So using hire purchase, you would pay your initial deposit of 5000 rounds. So off the 20,000 ran for the car, you really pay 5000 so you will still owe 15,000 rounds. Rice. Now the car company wants you to pay this off in 12 months. Each instalment on those 12 months is 1500 rounds. So let's have a look 12 times 1000 round. You're gonna end up paying 18,000 rounds for the car instead of 15,000 round. But you do get to do it every month and you're earning 4000 rounds per month. So you might be able to afford that amount even though it's quite a lot. So the higher purchase cost is equal to the deposit class. The total installments, right? So the cost of hire purchase is equal to the deposit which is at 5000 plus the total monthly instalments, and that is now 18,000. So what is the higher purchase costs 23,000 rand. So the car was only worth 20,000 rounds. But for you to have the advantage of allowing it right away, you paid 22,000 round after 12 months, so you might decide that this is worth it for the car. So that extra paid is your hire purchase cost minus the cash price. When we said cash price, it doesn't mean you pay for it in cash. It's something means you pay for it up front all at once. So the high approach or the extra paid is your 23,000 round miles or 20,000 and you can see that your extra paid was 3000 rounds. Let's have a look at another example. A DVD pair cost 480 round. It is available on hire purchase at a 20% deposit, followed by six installments off 75 grand. Each find the extra price. Okay, its first figure out. What that 20% of positives. So the deposit is equal to the cost price times the percent deposit. The cost price was 480 ran times 20%. That's 480. You know that we can just move the decimal twice on. We get tons 0.2 and that's 96 rounds. Okay, so that's what our initial deposit is. A total installments are the monthly installment of 75 round times a number of months, and then it's six. So that is equal 75 times six, and that is equal to 450 rounds. So what is the total hire purchase cost? That is the deposit of R 96 round pass up total monthly instalments of 450 round. So that is 96 plus 415 that is 546 rounds, and that extra paid is equal to the higher purchase cost minus the cash prize. So that is 556 running that we have you minus the cash price of 480 red, which is this over here. This is the extra we're paying on hire purchase, and that is 66 round. Great. So what I want to you to known hire purchase is the higher purchase cost is equal to the deposits past the total installments. The extra paid is equal to the higher purchase cost. Which is that over there, minus the cash price or the price of this selling for in the shop. If you can pay all at once. Well, then now we're gonna move on to loans and see how similar these two are. But what are the key differences?