Comparing fractions with like numerators and denominators

Let's compare the fraction 4/7 to the fraction 5/7. And what I want you to do right now is to pause this video and think about which of these fractions represents a larger quantity. I'm assuming you've had a go at it, and the one thing that might jump out at you is that they both have the same bottom number, which we call the denominator. They both have a denominator of 7. So one way to think about it is this is literally 4 sevenths. This is literally 5 sevenths. So we could rewrite 4/7 as literally 4 times 1/7. And we can rewrite 5/7 as literally 5 times 1/7. It's 5/7. So now if I have 4 of something versus 5 of something, which is going to be a larger quantity? Well, clearly 5 of this 1/7 is going to be more. So 4/7 is smaller, 5/7 is larger. And so what we can do is we can write a less than symbol. The way that I remember less than and greater than, is that the point, the small side of the symbol, always is on the same side as the smaller number. So this could be read as 4/7 is less than 5/7, or that 4 times 1/7 is less than 5 times 1/7. Now, let's do another scenario, but instead of having the same denominator, let's have the same numerator. So let's say we want to compare 3/4 versus-- let's say versus 3/9. Which of these two fractions is a larger number? And once again, pause the video and try to think about it on your own. Well, as we mentioned, we don't have the same denominator here. We have the same top number instead. Here is the same bottom number, same denominator. Here we're going to have the same numerator. We have the 3 right over there. And we could view 3/4 as literally 3 times 1/4. And we could view 3/9 as literally 3 times 1/9. So we have 3/4 and we have 3/9. So we really just have to about what's larger, a fourth or a ninth? Well, think about if you start with a whole-- think about starting with a whole like this. And let me make a whole right over here, so the same sized whole. A fourth is literally taking the whole and dividing it into 4 pieces. While a ninth is taking the whole and dividing it into 9. 9 equal sections, I could say. So let's divide this into 4 equal sections. So my best attempt to hand draw equal sections. So that's 2 equal sections and then that looks pretty close to 4 equal sections. So that right over there is 1/4. And let me draw ninths here. So let me first split this into 3 equal sections. So those would be third. And then split each of those into 3 equal sections. So this is my best attempt at that, at hand drawing 9 equal sections, splitting the whole into 9 equal sections. So when you see here-- and you might have already realized this-- if you divide something into 4 equal sections, each section is going to be bigger than if you divide it into 9 equal sections. A ninth is smaller than a fourth. A ninth is smaller than 1/4. So 3/9 is going to be smaller than 3/4. So once again, when you want to do less than or greater than, you want to put the point, the small side of the symbol, on the same side as the smaller number. So it would look like this. And this is the greater than symbol, because what you have on the left is the larger number. 3/4 is greater than 3/9. And if you actually wanted to represent not just 1/4 and 1/9, we could actually color it in. 3 times one fourth, well that's 1, 2, 3. While 3 times 1/9, or 3/9, is 1, 2, 3. And when you look at it that way, it's pretty clear. But the important thing to realize is, is that when the denominator is larger, you're dividing the whole into more pieces so each piece will be smaller. So making the denominator larger makes the fraction smaller. Making the numerator larger makes the fraction larger.