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'll 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 so you can try to estimate them but what you can get a very you'll be able to directly compare them if you 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 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 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 our least common multiple is going to have at least 1 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 12 and 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 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 1 3 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 1 3 right over there and then finally we need to be divisible by 15th 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 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 36 times 5 you could do that in your head if you 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 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 let me do it this way let me so if we do 4 over 9 to get the denominator of 9 to be 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 over 20 and so 4/9 is going to be the same thing as 80 80 over 180 now let's do the same thing for 3/4 3 over 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 4 times 45 4 times 40 is 160 4 times 5 is 20 Adam 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 135 135 and the denominator here is 100 180 now let's do four-fifths 4 over 5 to get 5 to get to our denominator to be 180 what you have to multiply 5 by so 5 let's see if you multiply 5 by 30 you'll have you'll get to 150 but you have to then you have another 30 so you have to multiple actually we know it right over here after x 36 x 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 over 180 and then we have only two more to do so we have our 11 12 11 over 12 is so to get the denominator to be 180 we have to multiply 12 by so 12 times 10 is 120 and then you have 60 left so they have to multiply it by 15 15 in the denominator and 15 in the numerator and so this gives us the denominator gives us 180 and 11 times 15 so 10 times 15 is 150 and then you have one more or 15 so it's going to be 165 165 and then finally we have 13 15 we have 13 13 15 to get to our denominator to be 180 you have to multiply it by 12 we already figured out that 12 times 15 is 180 so 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 put one more 12 than there you get 156 then do that right 12 plus 144 is going to be 156 so we've rewrite written each of these each of these fractions with a new 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 over 180 let me write it both ways 80 over 180 then the next smallest number looks like it's this 135 right over here so then our next one is going to be the 135 I want to do in that same color the next one is going to be that 135 over 180 which is the same thing as 3/4 which is the same thing as 3/4 and then the next one is going to be let's see we have the 144 over 180 so this is going to be the 144 over 180 which is the same thing as 4/5 and then finally where we have two more the next is this 156 over 180 so then we have our 156 over 180 which is the same thing as 13 15 13 over 15 and then we have one left over the 165 over 180 which is the same thing so we went do that in yellow we have our 165 over 180 which is the same thing as 11 12 which is the same thing as 11 12 and we're done we have finished we have finished our ordering so this is if you're doing the Khan Academy module on this you would this is what you would input into into that little box there