Comparing fractions 2 (unlike denominators) | Fractions | Class 6 | Mathematics | Khan Academy

Added: 2026-05-07

[00:00:00]Use less than, greater than, or equal to to compare the two fractions. 21 28s or 21 / 28 and 6 9ths 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 denominators. 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 / 7 is 3. And 28 / 7 is 4. So 21 28s is the exact same fraction as 3/4s. 3/4s is the simplified version of it. Let's do the same thing for 6 9ths. 6 and 9 are both divisible by 3. So let's divide them both by 3 so we can simplify this fraction. 6 / 3 is 2 and 9 / 3 is 3. So 21 / 28 is 3/4s. They're the exact same fraction just written a different way.

[00:01:36]This is the more simplified version. And 6 9ths is the exact same fraction as 2/3. So we really can compare 3/4s and 2/3. So this is really comparing 3/4s 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.

[00:01:58]Then we would have to multiply big numbers. Here we could do fairly small numbers. The common denominator of 3 over 4 and 2 over 3 is going to be the least common multiple of 4 and 3. and four and three 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 over 4 as something over 12. And we can write 2 over 3 as something over 12. And I got the 12 by multiplying 3 * 4. They have no common factors. Another way you could think about it is four. If you do a prime factorization is 2 * 2 and 3. It's already a prime number. So you can't prime factoriize it anymore.

[00:02:40]So, what you want to do is think of a number that has all of the prime factors of four and 3. So, it needs 1 2 another two and a three. Well, 2 * 2 * 3 is 12.

[00:02:50]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 * 3 is 9. Over here to get from 3 to 12, we have to multiply the denominator by 4.

[00:03:18]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 92. And 6 over 9 is the exact same thing as 8 over 12. So which of these is a greater quantity?

[00:03:37]Well, clearly we have the same denominator. Right? Now we have 9 / 12 is clearly greater than 82. So 92 is clearly greater than 82. Or if you go back and you realize that 92 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 over 9 is definitely greater than 6 over 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. So 21 over 28 and 6 over 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 * 14 and 14 is 2 * 7. That's its prime factorization. Prime factorization of 9 is 3 * 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 * 9. So let's over here multiply 28 * 9. There's a couple of ways you could do it. You could multiply in your head 28 * 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 * 8 is 72. 9 * 2 is 18. 18 + 7 is 25. So, we get 252. So I'm saying the 252 common denominator here is going to be 252. Least common multiple of 28 and 9.

[00:05:27]Well, to go from 28 to 252, we had to multiply it by 9. We had to multiply 28 * 9. So we're multiplying 28 * 9. So we also have to multiply the numerator * 9.

[00:05:40]So what is 21 * 9? That's easier to do in your head. 20 * 9 is 180. and then 1 * 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 * 28 6 * 20 is 120. 6 * 8 is 48. So we get 168. Let me write that out just to make sure I didn't make a mistake. So 28 * 6 8 * 6 is 48. 2 * 6 is 12 + 4 is 16. So right 168.

[00:06:28]So now we have a common denominator here. And so we can really just compare the numerators. And 189 is clearly greater than 168. 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 which is clearly greater than the right hand side which is really 6 over