How to Simplify Fractions

What does it mean to simplify fractions? Is it the same as reducing fractions? And is it actually important for you to be able to simplify fractions? (Hint: Yes, it is. Why? Read on to find out!)

Jason Marshall, PhD,
November 16, 2012
Episode #129

Given all the food-related images that fractions conjure up (pizzas and pies anybody?), there’s no doubt that they hold the distinction of being the topic in math that can best get your stomach rumbling. Of course, for many people, the sight of a few fractions is anything but appetizing. There are lots of things about working with fractions that can be tricky and perhaps even a little nauseating. But the truth is that these things really don’t have to drive you crazy. Which is why we’re going to spend the next few weeks talking about some important fraction fundamentals that will help you regain your appetite for these most delicious of mathematical treats. Up first today: How to simplify (aka, reduce) fractions.

What Is Simplifying Fractions?

What does it mean to simplify a fraction? Well, it just means that you rewrite the fraction in its simplest possible form. Of course, this now requires us to define what we mean by “simple”…which we’ll get to shortly. But since this whole thing can feel kind of circular, let's start by grounding ourselves with a classic real world culinary example that only fractions can offer. That’s right, let’s talk about pizza. If you take 1/4 of a pizza and put it together with another 1/4 of a pizza, what do you get? Well, the math tells us that after sticking two-quarters of a pizza together, you’ll have 1/4 + 1/4 = 2/4 of a pizza.

Of course, without even thinking about the math, most people who have ever eaten a pizza (or a portion thereof) will know that putting two-quarters of a pizza together will give them 1/2 of a pizza. From which we can conclude (I know this probably won’t be a big surprise to most of you) that the fractions 2/4 and 1/2 must represent the same total amount of pizza. Keep in mind that 2/4 and 1/2 don't actually represent the exact same thing in this case. After all, 2/4 means two slices that are each 1/4 of the total pizza and 1/2 means one slice that is 1/2 of the total pizza. While not being exactly the same thing in terms of slices of pizza, these two fractions nonetheless represent the same total quantity of pizza. So, in that sense, the two fractions 2/4 and 1/2 are equivalent.

What Does "Simple" Mean?

Which fraction is simpler: 2/4 or 1/2? Well, a fraction is said to be fully simplified when its numerator and denominator are reduced as much as possible. Since the numerator and denominator in 1/2 are both smaller (in other words, more reduced) than the numerator and denominator in 2/4, convention says that the fraction 1/2 is a simplified form of the fraction 2/4.

How to Simplify Fractions

Okay, how do you actually simplify fractions? The easiest way is to start by factoring the numerator and denominator of the fraction you’re trying to simplify. For example, to simplify the fraction 12/15, start by performing a prime factorization of the numbers 12 and 15. Using what we’ve learned about factoring numbers, we find that 12 can be factored into the trio of prime numbers 2, 2, and 3 (since 12 = 2 x 2 x 3), and 15 can be factored into the pair of prime numbers 3 and 5 (since 15 = 3 x 5).

Once you've performed these prime factorizations, the next step is to rewrite the fraction in terms of the prime factors. In our example, the fraction 12/15 can be rewritten as (2 x 2 x 3) / (3 x 5). The final step is to look for factors that appear in both the numerator and denominator—these are called “common factors”—and then, since dividing the top and bottom of a fraction by the same number doesn't change the value of its ratio, to divide both the numerator and denominator by any common factors we find. In our example of 12/15 = (2 x 2 x 3) / (3 x 5), we can divide the top and bottom by the common factor 3. When we do so, we find that 12/15 = (2 x 2) / 5 or 4/5. So the fraction 12/15 can be simplified or reduced to become the fraction 4/5.

Why is Simplifying Important?

And that's really all there is to simplifying fractions. Just remember to start by factoring the numerator and denominator, then rewrite the fraction in terms of these prime factors, next “cancel out” any common factors (which is effectively what happens when you divide the top and bottom of the fraction by the common factors), and finally multiply any remaining factors together to find the reduced fraction.

Why do you need to know how to simplify fractions? Is it actually important? In truth, in one sense it's not. Really?! Yes, really. After all, one could argue that simplifying fractions isn't a critical part of solving problems since the ratio represented by the reduced fraction is equivalent to the one you started with. All reducing fractions does is clean things up and simplify your life. But hang on…those are both really, really good things! And they're precisely the reasons that you and everybody else should learn to simplify all the fractions in your life.

Wrap Up

Okay, that's all the math we have time for today. Remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too. Finally, please send your math questions my way via Facebook, Twitter, or email at mathdude@quickanddirtytips.com.

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!

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