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Comparing fractions 2 (unlike denominators)
Use less than, greater than, or equal to compare the two fractions 21/28, or 21 over 28, and 6/9, or 6 over 9. So there's a bunch of ways to do this. The easiest way is if they had the same denominator, you could just compare the numerators. Unlucky for us, we do not have the same denominator. So what 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 feeling that'll be the fastest way to do it. So 21/28-- 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 the simplified version of it. Let's do the same thing for 6/9. 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 21/28 is 3/4. They're the exact same fraction, just written a different way. This is the more simplified version. And 6/9 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 could 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 prime factors with each other. So their 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 4, if you do a prime factorization, is 2 times 2. And 3-- it's already a prime number, so you can't prime factorize it any more. So what you want to do is think of a number that has all of the prime factors of 4 and 3. So it needs one 2, another 2, and a 3. Well, 2 times 2 times 3 is 12. And either way you think about it, that's how you would get the least common multiple or the common denominator for 4 and 3. Well, 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 8. And so now when we compare the fractions, it's pretty straightforward. 21/28 is the exact same thing as 9/12, and 6/9 is the exact same thing as 8/12. So which of these is a greater quantity? Well, clearly, we have the same denominator right now. We have 9/12 is clearly greater than 8/12. So 9/12 is clearly greater than 8/12. Or if you go back and you realize that 9/12 is the exact same thing as 21/28, we could say 21/28 is definitely greater than-- and 8/12 is the same thing as 6/9-- is definitely greater than 6/9. 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 didn't think to simplify our two numbers first. I'm trying to find a color I haven't used yet. So 21/28 and 6/9. So we could just find a least common multiple in the traditional way without simplifying first. So 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 over here multiply 28 times 9. There's a couple of ways you could do it. You could multiply in your head 28 times 10, which would be 280, and then subtract 28 from that, which would be what? 252. Or we could just multiply it out if that confuses you. So let's just do 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 common denominator here is going to be 252. Least common multiple of 28 and 9. Well, to go from 28 to 252, we had to multiply it by 9. We had to multiply 28 times 9. So we're multiplying 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 mistake. So 28 times 6-- 8 times 6 is 48. 2 times 6 is 12, plus 4 is 16. So 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 189/252 is clearly greater than 168/252. Or that's the same thing as saying 21/28, because that's what this is over here. The left-hand side is 21/28, is clearly greater than the right-hand side, which is really 6/9.