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Comparing and ordering fractions
What I want to do in this video is order these fractions from least to greatest. And the easiest way and the way that I think we can be sure we'll get the right answer here is to find a common denominator, because if we don't find a common denominator, these fractions are really hard to compare. 4/9 versus 3/4 versus 4/5, 11/12, 13/15. You can try to estimate them, but you'll be able to directly compare them if they all had the same denominator. So the trick here, or at least the first trick here, is to try to find that common denominator. And there's many ways to do it. You could just pick one of these numbers and keep taking its multiples and find that multiple that is divisible by all the rest. Another way to do it is look at the prime factorization of each of these numbers. And then the least common multiple of them will have to have at least all of those prime numbers in it. It has to be composed of all of these numbers. So let's do it that second way. And then let's verify that it definitely is divisible. So 9 is the same thing as 3 times 3. So our least common multiple is going to have at least one 3 times 3 in it. And then 4 is the same thing as 2 times 2. So we're going to also have to have a 2 times 2 in our prime factorization of our least common multiple. 5 is a prime number. So we're going to need to have a 5 in there. And then 12-- I'm going to do that in yellow. 12 is the same thing as 2 times 6, which is the same thing as 2 times 3. And so in our least common multiple, we have to have two 2's. But we already have two 2's right over here from our 4. And we already have one 3 right over here. Another way to think about it is something that is divisible by both 9 and 4 is going to be divisible by 12, because you're going to have the two 2's. And you're going to have that one 3 right over there. And then, finally, we need to be divisible by 15's prime factors. So let's look at 15's prime factors. 15 is the same thing as 3 times 5. So once again, this number right over here already has a 3 in it. And it already has a 5 in it. So we're cool for 15, for 12, and, obviously, for the rest of them. So this is our least common multiple. And we can just take this product. And so this is going to be equal to 3 times 3 is 9. 9 times 2 is 18. 18 times 2 is 36. 36 times 5, you could do that in your head if you're like. But I'll do it on the side just in case. 36 times 5, just so that we don't mess up. 6 times 5 is 30. 3 times 5 is 15 plus 3 is 180. So our least common multiple is 180. So we want to rewrite all of these fractions with 180 in the denominator. So this first fraction, 4/9, is what over 180? To go from 9 to 180, we have to multiply the denominator by 20. So let me do it this way. So if we do 4/9, to get the denominator of 9 to be 180, you have to multiply it by 20. And since we don't want to change the value of the fraction, we should also multiply the 4 by 20. So we're just really multiplying by 20/20. And so 4/9 is going to be the same thing as 80/180. Now, let's do the same thing for 3/4. Well, what do we have to multiply the denominator by to get us to 180? So it looks like 45. You could divide 4 into 180 to figure that out. But if you take 4 times 45, 4 times 40 is 160. 4 times 5 is 20. You add them up. You get 180. So if you multiply the denominator by 45, you also have to multiply the numerator by 45. 3 times 45 is 120 plus 15. So it's 135. And the denominator here is 180. Now, let's do 4/5. To get our denominator to be 180, what do you have to multiply 5 by? Let's see. If you multiply 5 by 30, you'll get to 150. But then you have another 30. Actually, we know it right over here. You have to multiply it by 36. Well, then you have to multiply the numerator by 36 as well. And so our denominator is going to be 180. Our numerator, 4 times 30 is 120. 4 times 6 is 24. So it's 144/180. And then we have only two more to do. So we have our 11/12. So to get the denominator to be 180, we have to multiply 12 by-- so 12 times 10 is 120. Then you have 60 left. So you have to multiply it by 15, 15 In the denominator, and 15 in the numerator. And so the denominator gives us 180. And 11 times 15. So 10 times 15 is 150. And then you have one more 15. So it's going to be 165. And then, finally, we have 13/15. To get our denominator to be 180, have to multiply it by 12. We already figured out that 12 times 15 is 180. So you have to multiply it by 12. That will give us 180 in the denominator. And so you have to also multiply the numerator by 12, so that we don't change the value of the fraction. We know 12 times 12 is 144. You could put one more 12 in there. You get 156. Did I do that right? 12 plus 144 is going to be 156. So we've rewritten each of these fractions with that new common denominator of 180. And now, it's very easy to compare them. You really just have to look at the numerators. So the smallest of the numerators is this 80 right over here. So 4/9 is the smallest. 4/9 is the least of these numbers. So let me just write it over here. So this is our ordering. We have 4/9 comes first, which is the same thing as 80/180. Let me write it both ways-- 80/180. Then the next the smallest number looks like it's this 135 right over here. I want to do it in that same color. The next one is going to be that 135/180, which is the same thing as 3/4. And then the next one is going to be-- let's see, we have the 144/180. So this is going to be the 144/180, which is the same thing as 4/5. And then we have two more. The next is this 156/180. So then we have our 156/180, which is the same thing as 13/15. And then we have one left over, the 165/180, which is the same thing-- I want to do that in yellow. We have our 165/180, which is the same thing as 11/12. And we're done. We have finished our ordering. So if you're doing the Khan Academy module on this, this is what you would input into that little box there.