How to Divide Fractions Using Invert and Multiply

Learn what it really means to divide two fractions and a trick you can use to make dividing fractions easier.

Jason Marshall, PhD,
November 5, 2010
Episode #040

A few months ago we talked about how to multiply fractions. Now that we’re comfortable doing this and we’ve talked about a few applications of multiplying fractions such as how to convert units and how to estimate how fast someone is running, the next logical thing to do is learn how to divide fractions too. So, that’s exactly what we’re going to talk about today.

How to Divide One Integer by Another

But before we get to fractions, let’s start by talking about how to divide one integer by another. Now, you might be wondering: Isn’t that just normal division? It’s true, it is. But my goal isn’t for us to talk about the mechanics of doing normal division, but rather to review what it really means. So then what does a problem like 6 divided by 2 really mean? Well, it’s just another way to ask the question: “How many times does 2 go into 6?” And, of course, the answer is 3. You can think of it this way: Imagine you have 6 apples which you then divide up into groups of 2 apples. That means you have 3 groups of 2 apples in front of you—so 6 divided by 2 (or 6 divided into groups of 2) equals 3. Yes, I know this is an extremely simple example, but having a simple example in mind will help as we move to tougher topics.

Okay, before moving on, I want to remind you about something we talked about in the article on how fractions and division are related that’s going to be really important for understanding today’s topic. What is it? It’s the idea that dividing a number by 2 is the same as multiplying it by the fraction 1/2. So, the division problem 6 divided by 2, is equivalent to the multiplication problem 6 times 1/2. That means, rather interestingly, that a problem of dividing integers can be turned into a problem of multiplying fractions. And that’s going to come in very handy in a few minutes.

How to Divide a Fraction by an Integer

Now, let’s step up the complexity ladder one rung. Instead of dividing an integer by another integer, let’s divide a fraction by an integer. Take the problem 1/2 divided by 3, for example. What does that really mean? Well, it’s asking: “How many times does 3 fit into 1/2?” Right off the bat we know that the answer has to be a number smaller than 1, since 3 doesn’t fit into 1/2 any whole number of times. But it will fit into 1/2 some fractional number of times. What fraction? Well, let’s go back to using the relationship between fractions and division which tells us that the problem 1/2 divided by 3 is equivalent to the problem 1/2 times 1/3—which equals 1/6. And that means that we’ve once again turned a division problem back into a problem of multiplying fractions.

How to Divide an Integer by a Fraction

Okay, let’s turn the problem of dividing a fraction by an integer on its head and instead talk about dividing an integer by a fraction. How about the problem 2 divided by 1/4? What does it really mean? Well, this is where things start to get a bit tougher. The problem 2 divided by 1/4 is asking how many times 1/4 will go into 2. You can think of it this way: Imagine you have two oranges which you divide up into quarters. The question is then: How many of those quarter wedges will fit into 2 oranges? Of course, the answer must be 8 because each orange has 4 quarters, and there are 2 oranges—so 4 times 2 equals 8.

That wasn’t too bad, right? But it’s not always so easy. Here’s what I mean: What if the problem wasn’t 2 divided by 1/4, but was something harder like 7 divided by 8/9 instead. Then you’d be left trying to figure out how many times 8/9 goes into 7—and that’s definitely tougher to do in your head! There’s got to be a better way. And there is. So, what’s the trick?

Divide Fractions Using Invert and Multiply

The quick and dirty tip to make dividing fractions easier is to remember to invert and multiply. Here’s what it means. Let’s go back to our problem 2 divided by 1/4, and let’s think of this as a big fraction with 2 in the numerator and the fraction 1/4 in the denominator. The invert part of “invert and multiply” means to take the denominator of this big fraction, 1/4, and invert it. In other words, flip it on its head so its numerator becomes its denominator and vice versa. The inverse of 1/4 is therefore 4/1, or just 4. Now for the multiply part of “invert and multiply”: all you need to do is multiply the 2 from the initial problem by the inverted denominator, 4. So, that’s 2 times 4, which of course equals 8—just like we calculated earlier. But, unlike how we calculated this earlier, we now have an easy method for doing harder problems too. Take 7 divided by 8/9. All we have to do is invert 8/9 to get 9/8, and then multiply this by 7 (numerator: 7 x 9 = 63; denominator: 1 x 8 = 8) to find that the answer is 63/8, or 7 and 7/8.

Why Does Invert and Multiply Work

[[AdMiddle]But why does “invert and multiply” work? Well, it works for the exact same reason that we were able to turn the problem of dividing two integers into a problem of multiplying fractions way back at the beginning of this article. In other words, when we turned 6 divided by 2 into the problem 6 times 1/2, we were just inverting and multiplying. And each of the reasons we talked about why it worked then are still valid for these other types of problems—be it dividing an integer by a fraction or even dividing a fraction by a fraction. Just remember to invert and multiply and your life with fractions will be much easier. Of course, don’t forget why it works too—it’s always a good idea to understand how and why your tools do what they do before you go around trying to use them…that way you don’t try to use something like a sledgehammer for hanging a picture.

Wrap Up

Okay, that’s all the math we have time for today. Please email your math questions and comments to mathdude@quickanddirtytips.com. You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.

Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!