Comparing fractions with same numerator | Math | 3rd grade | Khan Academy

Let's compare 5/6 and 5/8.

Let's think about what they mean. 5/6 means five out of six pieces. If you have a whole, let's say a whole cake, and you cut it into six pieces, 5/6 is five of those six pieces. 5/8 again is five pieces. That's something that's the same; we're both talking about five pieces, but this time, we've split our cake into eight pieces. So, it's five out of eight pieces.

We can represent that by drawing it. Maybe we can draw; here's one whole and then another one, so these are two equal wholes. On one of them, we can shade 5/6, and the other 5/8. That way, we can look and compare them.

For 5/6, if we divide this whole—we were using the example of cake—into six equal-sized pieces. Not sure if those are perfect, but let's say those are six equal-sized pieces. 5/6 is talking about five of those pieces, so one, two, three, four, five. This image represents 5/6.

Now for 5/8, let's think about it for a second. Will the pieces in 5/8 be bigger or smaller than 5/6? Are eights bigger or smaller than six? Well, we can draw and see if we have the same size whole, which we need to have the same size whole, and we draw and we split it into eight pieces this time. So now we have fourths, and these pieces, these 1/8, are smaller than a sixth, because this time we had to split the cake between eight people, so we got smaller pieces.

Now again, we can shade five of them. It seems like five pieces might be equal to five pieces, but we can look and see. Four, five. Now we can look and see that 5/6 takes up more space, takes up more area. This is a larger amount, and the reason is because each piece is larger.

So five larger pieces is more than five smaller pieces, or is greater than. And remember, with our symbol, the open side, the larger side, should be facing our larger number. So 5/6 is greater than 5/8.

Here's one more, but this time let's try to compare them without drawing. Let's just think about what these mean. 2/5 means two pieces out of five. So one whole was split five ways and we got two of the pieces. 2/3 means that same whole was split three ways and we got two of the pieces.

Well, which two pieces are larger? The 2/3 are larger, 'cause if we only have to split our pieces three ways, we can have larger pieces. So the two-fifths are smaller. These two smaller pieces are less. These two smaller pieces are less than the two larger pieces.

And again, the open end of our symbol should be facing the larger number. So 2/5 is less than 2/3.