Question

How to reduce fraction difficult numbers

Answer 1

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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.