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# How to Add and Subtract Mixed Fractions

What’s 2-3/7 + 4-2/9? How about 4-2/9 – 2-3/7? How can you add or subtract any pair of mixed fractions? Keep on reading to find out!

By
Jason Marshall, PhD,
January 4, 2013
Episode #136

Now that we’ve tackled common denominators, least common multiples, lowest common denominators, and converting between mixed and improper fractions, we’re ready to take a look at some real world problems that use these ideas. In particular, today we’re going to learn how to add and subtract mixed fractions. What exactly does that mean? Well, imagine you’re measuring the length of something that’s made of two pieces. One piece is 2-3/8 inches long and the other is 4-11/16 inches long. The questions is: What’s the total length? Keep on reading because that’s precisely the problem we’ll be solving today.

## A Word About “Real World Math”

Sometimes math looks really abstract and feels like it’s living in some alternative universe from the one that your day-to-day life takes place in. But it’s important to remember that math is actually all around you…and it really is what makes this universe tick. Just take a look at the stuff that’s surrounding you or think about the things you do every day and you’re sure to find math.

For example, the problem that I mentioned earlier about measuring the total length of something that’s made up of two or maybe even more parts certainly involves math…in particular, it involves mixed fractions. Since solving this problem is our goal for today’s show, let’s get things started with a brief review of some of the most important background ideas.

## Review of Adding and Subtracting Fractions

To solve our measurement problem, we need to learn how to add and subtract mixed fractions—like 2-1/2 and 3-1/3—that are made of a whole number and a fractional part. But in order to add and subtract mixed fractions, we need to know how to add and subtract regular fractions like 1/2 and 1/3 that don’t have whole number parts.

As we’ve learned, we can do that by rewriting all the fractions in the problem in terms of a common denominator—the lowest common denominator is often a good choice. To get the answer, we just add or subtract the numerators of the fractions and write the result over the common denominator. For example, since 1/2 and 1/3 can be rewritten 3/6 and 2/6 in terms of their lowest common denominator, 6, we can add them by adding their numerators 3 and 2 to get 5 and by then writing this sum over the common denominator 6. Which means that 1/2 + 1/3 = 3/6 + 2/6 = (3+2)/6 = 5/6.

After you’ve got this basic idea down, you should find that adding and subtracting regular—aka, proper and improper—fractions is actually pretty easy. But what about today’s main topic: mixed fractions? Well, thankfully, it turns out that they aren’t much harder.

## How to Add and Subtract Mixed Fractions — Method 1

So, how do you add and subtract mixed fractions? The best way is to start by turning all the mixed fractions in the problem into improper fractions. That’s the key because once you’ve done that you’ll have turned the problem into a problem that you already know how to solve. In other words, you’ll have turned the problem of adding or subtracting mixed fractions into a problem of adding or subtracting regular fractions! If you’re not quite sure how to convert mixed fractions into improper fractions, I encourage you to take a look at last week’s show on that topic before moving on.

To see how this all works, let’s take a look at our measurement example from before where we were trying to add the two lengths 2-3/8 inches and 4-11/16 inches. The first step is to convert these two mixed fractions into improper fractions. I’ll let you work out the details, but when you do this you should find that 2-3/8 = 19/8 and 4-11/16 = 75/16. All that’s left to do now is add the improper fractions 19/8 and 75/16. Rewriting 19/8 in terms of the lowest common denominator, 16, we find that 19/8 + 75/16 = 38/16 + 75/16 = (38+75)/16 = 113/16. If we want to, we can now convert this improper fraction back into a mixed fraction to find that 2-3/8 + 4-11/16 = 7-1/16.

How does subtraction work? Exactly the same way…except we subtract instead of add. For example, 4-11/16 – 2-3/8 = 75/16 – 38/16 = 37/16 or 2-5/16. That’s it!

## How to Add and Subtract Mixed Fractions — Method 2

But that’s not the only way to add and subtract mixed fractions. Instead of starting by converting the mixed fractions into improper fractions, we can instead use the fact that a mixed fraction is really a sum of a whole number and a proper fraction along with the associative property to do things a bit differently. In particular, since mixed fractions like 2-3/4 and 1-1/2 really just mean 2 + 3/4 and 1 + 1/2, a problem like 2-3/4 + 1-1/2 is actually the same as the problem 2 + 3/4 + 1 + 1/2.

Since the associative property of addition allows us to swap these numbers around and add them up in any order we’d like, we’re free to instead turn this into the problem (2 + 1) + (3/4 + 1/2). In other words, we can turn any addition or subtraction problem involving mixed fractions into two separate addition or subtraction problems—one for the whole number parts and one for the proper fraction parts. In this case, the whole number part is 2+1=3 and the proper fraction part is 3/4 + 1/2 = 5/4 = 1-1/4. To find the total sum, we just need to add these two parts together to find that 2-3/4 + 1-1/2 = 3 + 1-1/4 = 4-1/4.