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Multiply and Divide Fractions
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  1. Multiplying fractions -- examples 1, 2
  2. example 3
  3. Mixed numbers to Improper fractions
  4. example 4
  5. example 5
  6. Dividing Fractions -- examples 1, 2
  7. example 3
  8. example 4
  9. example 5
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(Instructor, Patrick Martin) Okay, we're going to study how to multiply and divide fractions. The easiest thing to do is, first of all, to multiply fractions. So, let's take notes, as usual. And, let me give you this question: 1/2 of 1/4 equals (and you fill in the answer) What do you think? So, what's a half of a quarter of an inch, for example? You can press pause if you need to. Pause is located in the bottom left of the screen. 1/2 of 1/4 inch? 1/8 of an inch, do you think? What we're gonna do is check the rule for multiplying fractions, and see if it works. "of" means multiply, funny enough. So, we can change this to 1/2 multiplied by 1/4 Now, the rule for multiplying fractions is: we multiply the tops (1 times 1) And then we multiply the bottoms (2 times 4). and we get, 1 over 8 (1/8), and that's correct. Okay? Let's try another one for fun: 2/3 of 15 equals? What do you think? Write down the answer. Just guess it. 2/3 of 15 dollars... So, you might want to think, what's 1/3 of $15? So, what's 2/3 of $15? And, again, if you want to take some time and think about it, press pause. Pause is located in the bottom left of the screen. $10 dollars, right? 1/3 is 5. 2/3 is 10. So, let's check There's a method for multiplying fractions. And here's how we do it, we go: 2/3 Now, of actually means multiply, in math. So, change that to multiply. And 15 can be written 15/1. And, if we multiply the tops (2 times 15), over and then multiply the bottoms (3 times 1), we get (2 times 15 is) 30. Over (3 times 1 is) 3. And, write down the answer. 30 divided by 3. 10, right? So, it does work. The rule for multiplying fractions is multiply the tops, then multiply the bottoms, and then divide. The tops are called the numerator. The top of a fraction is called the numerator, and the bottom is called the denominator. So, we multiply the numerators, and then we multiply the denominators, and then divide. So, the rule for multiplying fractions, and we can use letters to represent it: a over b (a/b) multiplied by c over d (c/d). How do you think we would write that? a over b times c over d? Well, we multiply the tops (a times c). And then multiply the bottoms (b times d). And, in algebra, when letters or variables are written beside each other that means multiply. So that's ac all over bd (ac/bd), okay? So, let's have a look at this then: 3/8 multiplied by 2/9. Get the answer. So, again, multiply the tops (3 times 2) over 8 times 9, multiply the bottoms. and we get.... 6, over 72 (6/72). Now, the question is: Can we put the fraction in lowest terms? So, just press pause (bottom left of the screen), and put that fraction in lowest terms. Now, if you think about it- well, in fact, 6 goes into 72.... 6 into 6 goes once, 6 into 7 goes once, remainder 1. 6 into 12, goes twice. So, 6/72, can be written 1/12, in lowest terms. Okay? Now, the thing about this example is that this was a lot of work to get to here, and we had to do a lot of computation multiplying on top and bottom, and then putting it in lowest terms. A quicker way is this way: If you're multiplying fractions, you can do a thing called "cross-cancelling". If you look at 3 and 9, they both have a common factor of 3. They're both a multiple of 3. 3 into 3 goes once. 3 into 9 goes 3 times. Can you see anything else we can do? How about 2 and 8? They're both multiples of 2. 2 into 2 goes once. 2 into 8 goes 4 times. And, so, a quicker way to multiply these fractions, is first to cross-cancel, and now we'll get 1 times 1 1 times 1 over, 4 times 3, and that makes 1 over.... 12. ( 1/12 ) Okay, so you try another one now. 4 over 15 multiplied by 35 over 12. 4/15 x 35/12 Okay, so, same thing. We can multiply the numerators, and then multiply the denominators, and divide. But, that looks like a lot of work. How about we cross-cancel first? Do 4 and 12 have a common factor? 4 goes into both, right? So, 4 into 4 goes once, 4 into 12 goes 3 times. How about 15 and 35? 5 into 15, goes 3 times. 5 into 35, goes 7 times. Right? So, we end up with 1 times 7, divided by 3 times 3, which is 7/9. Okay? So, now let's review changing mixed numbers to improper factions before we proceed. So, for example: 2 3/4 Change that to an improper faction. Right now, it's a mixed number. Changed it to an improper fraction. So, let's think about it. $2.00 and ¾, what do you think? Well, there's 4 quarters in a dollar. So, this would be 8 quarters. Plus 3, that's 11, isn't it? 11 quarters. And here's the procedure: We go 4 times 2 (4⋅2), and then we add 3 and put it all over the 4. (4⋅2+3 / 4) So, that gives us 8 plus 3 over 4. 11 over 4. 11 quarters, right? So, you try this one then: 3 and 5/8 of an inch. How many eighths of an inch is that? 3 5/8 inch... Well, there's 8 eighths in 1 inch. So, 8 times 3 is 24. That's 24/8. Plus 5, that's 29 eighths, isn't it? And, here's the procedure: We go 8 times 3, then we add the 5. All over 8. 8 times 3, that's 24. Plus 5. All over 8. So, 29 over 8, right? (29 ∕ 8) So, if we were to multiply 3 and 3 fifths, by 2 and 7 ninths. The best way to do this, if you multiply 2 mixed numbers, the best way to do it is to, first, turn them both into improper fractions. So, 5 times 3, plus 3 over 5. That's 15, plus 3. (18 ∕ 5) Right? This one is 9 times 2. So, that's multiplied by, you know, 9 times 2, plus 7, all over 9. And that's 18, plus 7. 25 over 9. Now we multiply the fractions. And, if you want, you can multiply the numerators, and then multiply the denominators, and divide. But, if you can find factors that cross-cancel, do so. 9 into 9 goes once. 9 into 18 goes twice. Anything else? 5 into 5 goes once. 5 into 25 goes 5 times. So, this gives 2 times 5, over 1 times 1. or 10/1 and 10 divided by 1. How many ones are in 10? Well, there's 10 one's in 10. Let's just quickly review that. I mean, if you have 6 divided by 2, how many 2's are in 6? There's three 2's in 6, right? So, let's see: 20 divided by 4. How many 4's are in 20? Five 4's in 20. So, if you had 3/1 How many 1's are in 3? How many $1.00 bills in $3.00? There's 3 of them, right? So, let's do one more, for practice. 1 3/4 multiplied by 2 1/7 Multiplying two mixed numbers: 4 times 1 is 4, plus 3. That's 7 quarters. Multiplied by (7 times 2), that's 14. 14 plus 1, 15 sevenths. (15 ∕ 7) Now, cross-cancel and see what that gives. Well, we can do this: 7 into 7 goes once. 7 into 7 goes once. So, we have 1 times 15, over 4 times 1 Which gives, 15 over 4. (15 ∕ 4) Now, just for fun, this is an improper fraction. We usually use improper fractions in algebra, and you'll see why when we get to equations. Sometimes we use mixed numbers. So, let's turn it into a mixed number, for fun. 4 into 15 goes how many times? 4 into 15 goes.... 3 times, doesn't it? And the remainder is 3. Because it's 12, and so, 3 remainder 3. Or, that can be written: 3 and 3 quarters, right? (3 3 ∕ 4). Now, dividing fractions. You must have first covered multiplying fractions, because that's the skill we need most of all. There's only one extra step to dividing fractions. Let's first understand the concept. Write down the answer to this: 10 divided by 10? 10 ÷ 10? 10 divided by 5, 10 divided by 1. Okay? Don't be afraid to write it down. That's how you get things into your long-term memory, is to write it on a piece of paper. You remember 40% of what you write down. How many 10's are in 10? There's one 10. How many 5's are in 10? How many $5.00 bills in $10.00? Well, there's 2 of them. How $1.00 bills are in $10.00? 10 of them, right? Easy! Okay, well. how about this: 10 divided by 1/4? Write the answer down now. This is exactly the same as these ones... Okay? We need to understand something about division. Again, what division means is, how many 10's are contained in 10? There's one 10. And how many 5's are in 10? There's two. How many 1's are in 10? There's ten. How many quarters in $10.00? Write down the answer. Well, let's think about it. I mean, there's 4 quarters in $1.00 So, in $10.00 Press "pause" 40? That's correct. So, how about this one: 10 divided by ½ ? If you had 10 cups of sugar, to put into a big cake, how many half cups would that be? How many ½ cups of sugar in 10 cups of sugar? 10 divided by ½ ? Press "pause". Write down the answer. 20, isn't it? Okay? Now, let's look at our procedure for dividing by fractions. Because we have a procedure. 10 divided by 5. Now, in the previous video, we discussed how 10 can be written as 10 ∕ 1, right? 5 can be written as 5 ∕ 1. So, if we change these to fractions, we have 10/1 divided by 5/1. Now, the procedure for dividing by fractions is to get the reciprocal of the fraction on the right, and multiply. So, we do this. 10 over 1 multiplied by (and we flip this one upside down), 1 over 5, right? Now multiply the fractions: 10 times 1 is 10. Over 1 times 5 is five. 10/5 is 2. So, yes, 10/5 is definitely 2. But, what we've shown by this, is that this procedure, of dividing fractions, is true. When you flip the fraction on the right, and multiply. That is correct. So, let's explore that with 10 divided by ¼. If we change 10 to a fraction, it can be written as 10 over 1 . So, 10/1 divided by ¼ is the same as, (this is the procedure for dividing fractions) you go: 10/1 multiplied by the reciprocal of the fraction on the right. 4 over 1, right? (4/1) Now, that gives us 10 times 4, 40. Over 1 times 1, 1. And, 40 over 1 is How many 1's in 40? 40, right? And, we proved over here: 10 divided by ¼ is 40, when we thought about it. And now, we have also proven that the procedure for dividing by fractions, which is to flip this fraction on the right, and multiply. So, if you had, for example, 2 sevenths, divided by 3 fourteenths. All you've got to do is go: 2 over seven multiplied by, (flip the fraction on the right), 14 over 3. And, now multiply the fractions. And, of course, we can cross-cancel. 7 into 7 goes once. 7 into 14 goes twice. So, that gives us 2 times 2 is 4. Over 1 times 3 is 3. 4/3. So, how about this? If you had a over b, divided by c over d. What would that be? a, b, c, and d represent numbers. Well, that would be: the first fraction, a/b multiplied by (and what did we do with this fraction?) get the reciprocal: d/c, and multiply. So, we just flip this fraction upside down, and then multiply. And that becomes a times d over, b times c, right? Okay, how about mixed numbers? 1 1/4, divided by 2 3/8 Well, dividing mixed numbers, you might want to change them to improper fractions, to begin with, right? 4 times 1 is 4. Plus 1, is 5. So, we go 4 times 1, plus 1. That's 5 over 4. So, we get 5/4, divided by 8 times 2, plus 3 over 8. Which is (16+3) 19 over 8 (19/8). And, when dividing fractions, we flip the fraction on the the right and multiply. So, 5/4 multiplied by 8/9, right? Go ahead and do it then. You know, at any stage of these videos, you'll want to be pressing "Pause" to see if you can get the next step, obviously. 4 into 4 goes once. 4 into 8 goes twice. And we have (5 times 2 is) 10, over (1 times 9 is) 9. 10/9 Of course, we could have changed these both to mixed numbers, as well. Just for fun. 3 into 4 goes once, remainder 1. So 1 1/3, right? 9 into 10 goes once, remainder 1. So, this is 1 1/9. Okay? So, you go ahead and do this one then. See if you've learned your lesson. 2 3/5 divided by 1 1/10. So, press "Pause", and calculate that. So, again, first of all, we've got to change them into improper fractions. 5 times 2 is 10, plus 3, 13. So, that's 13/5. Divided by (1x10 is) 10, plus 1, all over ten. That's (10 plus 1) 11/10. So, that's 13/5, and this is the fraction we flip. The one on the right, not the one on the left. That's a common mistake. You always flip the fraction on the right. Multiplied by 10/11. And we can cross-cancel: 5 into 5 goes once. 5 into that goes twice. 13 times 2, 26. 1 times 11, 11. So, 26/11 is the improper fraction. As a mixed number, 11 into that, goes twice. Remainder.... well, that's 22. So, remainder 4, right? so, 2 4/11.