Comparing improper fractions and mixed numbers

I've got pairs of mixed numbers and improper fractions, and I want to think about which of the two is larger. So 1 and 7/8, 39/10. So you could do this in your head. You could say 10 goes into 39, I'll even write it out, 10 goes into 39 3 times, 3 times 10. And you want to find the largest number of times 10 goes into this without going over. So you couldn't write a 4 over here, because then that would be 40. That would go over 39. 3 times 10 is 30. And then you have a remainder of 9. So you could rewrite this expression right over here. Instead of 39/10 you could write it as 30/10 plus 9/10. And 30/10 is just 3. So this is equal to 3 and 9/10. And you could do that in your head. You could say 10 goes into 39 3 times, and the remainder is 9. You have your 9/10. And that's essentially just doing this in your head. So now we can compare, and we can literally just look at the whole number parts. This is 1 and something, 1 and 7/8, and we're comparing it to essentially 3 and 9/10. 3 and 9/10 is clearly a larger number. We have a 3 out here instead of a 1, so we will write less than. And the way I always remember it is, the opening always faces the larger number. And the point is small. It always points to the smaller number. Now let's do this next one. 4 and 7/8 versus 49/9. So let's convert this to a mixed number. 9 goes into 49 5 times, and 5 times 9 is 45. So the remainder is going to be 4. The remainder is 4, so this is 5 and 4/9. Once again, we can literally just look at the whole number parts. 5 is clearly larger than 4, so once again, less than. Point facing the smaller number, opening facing the larger number. Now 2 and 1/2 versus 11/10. 10 goes into 11 only 1 time. And if you care about the remainder, it's 1. So it's 1 and 1/10. Which is clearly smaller than 2 and 1/2. You just look at the whole number parts. 2 is clearly larger than 1. So we want our opening of our less than or greater than sign to face the larger number. So we would write it like this. And this is greater than, so 2 and 1/2 is greater than 11/10. The little point facing the smaller number. 5 and 4/9 versus 40/7. 7 goes into 40, so let me rewrite this, 7 goes into 40 5 times. And then you're going to have a remainder of 5, because 7 times 5 is 35. You have a remainder of 5 to get to 40. So it's 5 and 5/7. And if that looks like I'm doing some type of voodoo, just remember, I'm really just breaking it up. I'm just really saying that 40/7 is the same thing as 35 plus 5/7. The largest multiple of 7 that is less than this number. And this is the same thing as 35/7 plus 5/7. And then this, 35/7 5. And 5/7 is just 5/7 there. This one is interesting because we have the same whole number out front on our mixed numbers. 5 versus 5. So now we actually do have to pay attention to the fractional part of our mixed number. We essentially have to compare 4/9 to 5/7. And there's a couple of ways to do this. You could get them to have the same denominator. That's probably the easiest way to do it. So you could rewrite-- so what's the least common multiple of 9 and 7? They share no factors, so really the least common multiple is going to be their product. So if we want to rewrite 4/9 we would write 63 in the denominator, that's 9 times 7. If we multiply the denominator by 7 we also have to multiply the numerator by 7. So that will be 28. Now 5/7, we're going to make the denominator 63. We're multiplying the denominator times 9. Then we have to multiply the numerator times 9 as well. 5 times 9 is 45. So here it's easy to see. 45/63 is clearly larger than 28/63. And so we could write this. And because the whole number of parts are the same, and 5/7 is the same thing as 45/63, and 4/9 is the same thing as 28/63, we can write that 5 and 4/9 is less than 40/7. Another way that you could have thought about 4/9 versus 5/7 is you could have said, well, how does 4/9 compare to 4/7? We have the same numerator. The denominator here is larger than the denominator here. But when you have a number in the denominator, the larger it is, the smaller the fraction. The smaller the absolute value of the fraction. So this right over here is a smaller quantity than 4/7. And 4/7 is clearly a smaller quantity than 5/7. So 4/9 is clearly smaller than 5/7, so we would have gotten the same result.