Until now, we have seen a bunch of examples of fractions. Fractions whose numerator is smaller than the denominator are called **proper fractions**. The point of proper fraction is that if you look at them on the number line, they all live lie on the number line between 0 and 1.

Let's revisit improper fractions for a moment. Because it can be hard to imagine just how much something like 354/22 is, we can represent it as a mixed number, which is a whole lot easier to wrap your head around. A mixed number is expressed as a whole number followed by a fraction. Who knows why the fraction is following the whole number—maybe it's lost. It is pretty maze-y in here.

For example, **.**

Since mathematicians like to abbreviate things, we leave out the addition symbol and just write .

To turn an improper fraction into a mixed number, let's think about the brownies again. Might as well get as much use out of them as possible before they go stale.

Write the fraction as a mixed number.

What this number is saying is that each pan of brownies is cut into 7 equal pieces, and we have 22 total pieces. That means we have 3 full trays of brownies (3 × 7 = 21), plus 1 single, lonely brownie left over.

Therefore, .

To turn an improper fraction into a mixed number without drawing pictures (although that *does* take all the fun out of it), we perform long division. Remember that a fraction *is* a division problem in which we just haven't done the actual dividing. However, if we do that here and take 22 divided by 7, using the remainder as the fractional part of our mixed number, we get the following:

Therefore, . What do you know - that's the same answer we got when we did it the other way! It's like we're caught in a wormhole of perfect logic.

We're sure you're dying to learn how to do this in reverse - how to turn a mixed number back into an improper fraction. As a magician, it's one thing to cut your assistant in half, but it's always nice to be able to put her back together again as well. We'll get to converting mixed numbers into fractions when we get to the section on adding fractions (after all, the mixed number is really just , or . However, if you've got ants in your pants and absolutely can't wait, see the section below for a sneak peak. If you're a sucker for suspense, you can skip it and come back after you've learned all about adding fractions...

Okay, here we go. To turn mixed numbers back into improper fractions, remember that mixed numbers are simply abbreviated addition with the plus sign surgically removed:

Since we're going to be speaking the language of "1/2"s, we want to translate everything into those terms. How many 1/2's does 6 equal? Remember last night when you overheard your mother swearing that she would only eat 1/2 of a chocolate chip cookie (hey, at least we're off brownies), and then she kept going back to the tin until she'd consumed 6 entire cookies? That was 12 halves, if you'll recall. If you don't recall, you can also get there this way:

You have 6 × 2 = 12 half-cookies plus another half-cookie, for a total of thirteen halves, or .

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