# Comparing Fractions

## Your Teacher May be a Complex Person, but She Likes Her Fractions Simplified

As you now know, there are infinitely many ways to express any particular fraction. Even more if you speak Spanish.

However, infinity is quite a lot. So, to keep your teacher from going batty attempting to decode your homework, be sure to reduce your fractions. That way, we're all speaking a common language. ¿Entiendes?

**Comparing Fractions**

Another solid skill we'll need is the ability to compare different fractions, even when their denominators aren't the same.

Pick any fraction you like. Actually, you look like you've got your hands full at the moment. We'll do it for you.

### Sample Problem

Now make the numerator bigger. You can use a bicycle pump, helium tank—whatever gets the job done.

Your new fraction is now larger than your original fraction, because you have more pieces of the same size. More pieces equals more size. That's why you never hear of any pirates looting for pieces of nine. Too heavy.

Now go back to your original fraction. This time, make the denominator bigger and keep the numerator the same size. Is your new fraction bigger or smaller than your original fraction?

The new fraction is smaller. Meaning we have the same number of pieces, but each piece is now smaller. What a rip-off. Good thing we held onto the receipt.

Try this with a couple other fractions to convince yourself of these rules:

1. If you keep the denominator the same and make the numerator bigger, the fraction gets bigger.

2. If you keep the numerator the same and make the denominator bigger, the fraction gets smaller.

If we make both the numerator *and* the denominator bigger, there's no telling what might happen. Everyone had better stand back, just in case. Could get ugly.

Because we don't deal with fractions like ^{5}/_{18}, ^{42}/_{53}, etc. on a daily basis, it can sometimes be a difficult task to compare one of these guys to another fraction and instantly be able to tell which is larger. That's why it's always nice to have matching denominators. They're like socks: when they don't match, it's going to make your life so much harder.

If we have two fractions with the same denominator, they're easy to compare. Whichever has the bigger numerator is the bigger fraction. So what do we do with a troublesome pair like ^{3}/_{5} and ^{2}/_{3}?

To compare them, let's think about the two Star Wars cakes you had at your last birthday.

### Sample Problem

Suppose that, after the party was over and the damage was done, ^{3}/_{5} of one cake remained, and ^{2}/_{3} of the other was left over. (Maybe you should've just gotten one cake.) Which partial cake is bigger?

The first cake was originally cut into fifths, so let's now cut the remaining 3 slices into thirds: ^{3}/_{5} × ^{3}/_{3} = ^{9}/_{15}.

With the second cake, which was originally cut into thirds, let's now cut the remaining 2 slices into fifths: ^{2}/_{3} × ^{5}/_{5} = ^{10}/_{15}.

So the 3 slices of the first cake become 9 smaller slices, and the 2 slices of the second cake become 10 smaller slices.

Now it's easy to compare the fractions. Each cake is now divided up into fifteenths. The first cake has 9 slices remaining, and the second cake has 10 slices remaining, with all slices being the exact same size. So:

That means is bigger. Not Jabba the Hutt bigger, but bigger nonetheless.

Although this method will always work when comparing two fractions, sometimes it isn't the most efficient way. Like when you don't have a couple of cakes and spatulas handy. Consider the following example.

### Sample Problem

We can multiply the top and bottom of the first fraction by 175 and the top and bottom of the second fraction by 200, but *yikes*. We'll have matching denominators, but at what cost? Our numbers are going to be massive, and while that's a good quality to have in a cruise ship, not so much in the "solving fraction problems" department. So if there's a way to break this down into smaller numbers somehow, that would be peachy keen.

Our other choice is to recognize that 200 = 8 × 25, and 175 = 7 × 25. The number 25 divides evenly into both of these suckers! So if we multiply the numerator and denominator of by 7 and multiply the numerator and denominator of by 8, the denominators will both be equal at 1400, and we'll be ready to compare the numerators.

We get 77 and 56, respectively, for our two numerators. Clearly, 77 is bigger, which means that is bigger than . How did we arrive at this conclusion? By using the idea of Least Common Denominators. Since every one of those three words starts with a capital letter, it *must* be important. So important that we hereby dedicate the next section to it.