Comparing unit fractions

So which of the following numbers is a greater: one third or one fifth? Pause this video and try to answer that all right.

Now let's think about this together, and the way that I can best think about it is by visualizing them. So let's imagine a hole. So this is a hole right over there, and then let's say that this is another hole right over there. I'm going to try to make these rectangles about looking about the same.

Now, how would I represent a third? Well, I would divide this hole into three equal sections. And so I'm going to try to divide it into three equal sections. So there are three equal sections right over there, or they're supposed to be three equal sections. These are hand drawn, so give me a little slack. But one of these three equal sections, well that's one third. So that is one one-third right over there.

Now what about one-fifth? Well then I would try to divide this into five equal sections. So one, two, three, four, and five equal sections. And so one-fifth would be just one of these fifths, so it would be that right over there.

So when you compare it like this, what's larger, one-third or one-fifth? And if it isn't obvious just yet, I could drag this one over so that we can compare them directly. You can see very clearly that one-third covers more of the whole. It's a larger fraction of the whole than one-fifth is. So one-third is greater than one-fifth.

And so you might have noticed an interesting pattern, or might start thinking about a pattern. You might have been tempted when you saw the five here. Five is larger than three, but one-fifth is less than one-third, or one-third is greater than one-fifth. And that is generally true: the larger the denominator, the smaller the fraction is going to be.

Why is that? Because you're dividing your whole into more equal chunks. So if you're only dividing to three—if it's one of three, or one-third of the hole, or if it's one of three equal chunks of the whole—it's going to be bigger than one of five equal chunks of the whole.

And so, based on this, how would you compare these two numbers? How would you compare two-thirds to two-fifths? Well, same idea here. A third is bigger than a fifth, so two-thirds is definitely going to be bigger than two-fifths. And you could see it here: two-thirds is that, while two-fifths is that right over there.

And I can do another example where I haven't even drawn it out. How would you compare 4 over 6 to 4 over 8? So 4/6 versus 4/8. Well, same idea: a sixth is larger than an eighth. One-sixth is greater than one-eighth because the denominator here is smaller. We have the same numerator, but the denominator is smaller. So four of the bigger things is going to be larger than four of the smaller things. So 4/6 is greater than 4/8.