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use less than greater than or equal to compare the two fractions 21:28 soar 21 over 28 and six nines or six over nine so there's a bunch of ways to do this the easiest way is if they had the same denominator you could could just compare the numerators unlucky for us we do not have the same denominator so we could do is we can find a common denominator for both of them and convert both of these fractions to have the same denominator and then compare the numerators or even more simply we could simplify them first and then try to do it so let me do that last one because I have a set I have a feeling that'll be the fastest way to do it so 21 over 28 you can see that they are both divisible by 7 so let's divide both the numerator and the denominator by 7 so we could divide 21 by 7 and we can divide so let me make the numerator and we can divide the denominator by 7 we're doing the same thing to the numerator and the denominator so we're not going to change the value of the fraction so 21 divided by 7 is 3 and 28 divided by 7 is 4 so 21 28 is the exact same fraction as 3/4 3/4 is a simplified version of it let's do the same thing for six ninths 6 and 9 are both divisible by 3 so let's divide them both by 3 so we can simplify this fraction so let's divide both of them by 3 6 divided by 3 is 2 and 9 divided by 3 is 3 so 21 over 28 is 3/4 they're the exact same fraction just written in a different way this is the more simplified version and six ninths is the exact same fraction as 2/3 so we really can compare 3/4 and 2/3 so this is really comparing 3/4 and 2/3 and the real benefit of doing this is now this is much easier to find a common denominator for than 28 and 9 then we would have to multiply big numbers here we can do fairly small numbers the common denominator of 3/4 and 2/3 is going to be the least common multiple of 4 and 3 and 4 and 3 don't share any any any prime factors with each other so they're least common multiple is really just going to be the product of the two so we can write 3/4 as something over 12 and we can write 2/3 as something over 12 and I got the 12 by multiplying 3 times 4 they have no common factors another way you could think about it is for if you do a prime factorization is 2 times 2 & 3 it's already a prime number so you can't prime factorize it anymore so what you want to do is think of a number that has all of the prime factors of 4 & 3 so it needs one - another 2 and a 3 well 2 times 2 times 3 is 12 either way you think about it that's how you would get the least common multiple or the common the common denominator for 4 & 3 well to get from 4 to 12 to get from 4 to 12 you've got to multiply by 3 so we're multiplying the denominator by 3 to get to 12 so we also have to multiply the numerator by 3 so 3 times 3 is 9 over here to get from 3 to 12 we have to multiply the denominator by 4 so we also have to multiply the numerator by 4 so we get so we get 8 and so now when we compare the fractions it's pretty straightforward 21 over 28 is the exact same thing as 9 12 and 6 9 is the exact same thing 6 9 is the exact same thing as 8 12 so which of these is a greater is a greater quantity well clearly we have the same denominator right now we have 9 twelfths is clearly greater than 8 12 so 9 12 is clearly greater than 8 12 or if you go back and you realize that 9 twelfths is the exact same thing as 21 over 28 we could say 21 over 28 is definitely greater than and 8 twelfths is that that same thing as 6 nights is definitely greater than six ninths and we are done another way we could have done it we didn't necessarily have to simplify that and let me show you that just for fun so if we were doing it with if we started with if we didn't think to simplify our our two numbers first trying to find a number a color I haven't used so 21 over 28 and 6 over 9 so we could just find a a least common multiple in the traditional way without simplifying first so what is what's the prime factorization of 28 it's 2 times 14 and 14 is 2 times 7 that's its prime factorization prime factorization of 9 is 3 times 3 so the least common multiple of 28 and 9 have to contain a 2 a 2 a 7 a 3 and a 3 or essentially it's going to be 28 times 9 so let's see here multiply 28 times 9 28 times 9 there's a couple of ways you could do it you could multiply it in your head 28 times 10 which would be 280 and then subtract 28 from that which would be like what 252 or we could just multiply it out if that confuses you so let's just do the the second way 9 times 8 is 72 9 times 2 is 18 18 plus 7 is 25 so we get 252 so I'm saying the the common denominator here is going to be two hundred and fifty to two hundred and fifty-two and 252 least common multiple of 28 and nine well to go from to go from 28 to 252 we had to multiply it by 9 we have to multiply 28 times 9 so we're multiplied 28 times 9 so we also have to multiply the numerator times 9 so what is 21 times 9 that's easier to do in your head 20 times 9 is 180 and then 1 times 9 is 9 so this is going to be 189 to go from 9 to 252 we had to multiply by 28 so we also have to multiply the numerator by 28 if we don't want to change the value of the fraction so 6 times 28 6 times 20 is 120 6 times 8 is 48 so we get 168 let me write that out just to make sure I didn't make a make a mistake so 28 times 6 is 8 times 6 8 times 6 is 48 48 2 times 6 is 12 plus 4 is 16 right 168 so now we have a common denominator here and so we can really just compare the numerators and 189 is clearly greater than 168 so 189 over 252 is create clearly greater than 168 over 252 or that's the same thing as saying 21 over 28 because that's what this is over here the left-hand side is 21 over 28 is clearly greater than the right-hand side which is really six ninths