# Using Prime Factorization to Reduce Fractions

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Having to reduce fractions is a common occurrence in mathematics and even in our daily lives. This lesson will define prime factorization and show how to use it to reduce fractions.

## Reducing Fractions

Imagine you are talking to a friend. Your friend is telling you about a really yummy fruit juice they make and they tell you that the juice is 8/10 fresh pineapple juice. Then, they pause and say, 'Well, actually the juice is 4/5 fresh pineapple juice.' What just happened there? How did the pineapple juice content go from 8/10 to 4/5? What happened is your friend reduced the fraction 8/10 to an equivalent fraction 4/5.

Reducing a fraction involves removing a common factor that both the numerator and denominator have in common, where a common factor is a number that divides into both the numerator and the denominator evenly. In the case of your friend, they mentally reduced 8/10 by recognizing that both the numerator and denominator have a factor of 2. On paper, the work would be as follows.

When we reduce a fraction, we create an equivalent fraction, or a fraction that has the same value. In other words, 8/10 and 4/5 have the same value. There are a lot of different ways to go about reducing fractions. We are going to look at a method that uses prime factorization.

## Prime Factorization

Prime factorization of a number involves factoring the number completely until all the factors are prime, where a prime number is a number that is only divisible by 1 and itself. For example, consider the number 56. To find the prime factorization of 56, we just start factoring and keep going until all the factors are prime numbers.

 56 = 7*8 7 is prime, factor 8 further 56 = 7*2*4 7 and 2 are prime, factor 4 further 56 = 7*2*2*2 All the factors are prime

We see the prime factorization of 56 is 7*2*2*2. What's really neat about prime factorization is that it doesn't matter what two factors you start with, you will still get the same prime factorization of that number. To illustrate this, suppose we started with 56 = 2*28 instead of 56 = 7*8.

 56 = 2*28 2 is prime, factor 28 further 56 = 2*7*4 2 and 7 are prime, factor 4 further 56 = 2*7*2*2 All the factors are prime.

The factors are in a different order, but that doesn't matter. We still get that the prime factorization of 56 is 7*2*2*2. Now let's look at how to use prime factorization to reduce fractions.

## Using Prime Factorization to Reduce Fractions

To use prime factorization to reduce fractions, we follow these steps.

1. Find the prime factorization of the numerator and the denominator, and rewrite the fraction with those prime factorizations.
2. Cancel out any common factors in the numerator and the denominator.
3. Multiply any factors that are left in the numerator together and any factors that are left in the denominator together. This is your fraction in reduced form.

Let's take a look at your friend's juice again. Your friend said that the juice is 8/10 pineapple juice. To reduce this fraction using prime factorization, we first find the prime factorization of the numerator and the denominator and rewrite the fraction.

 8 = 2*4 2 is prime, factor 4 further 8 = 2*2*2 All the factors are prime

The prime factorization of 8 is 2*2*2. As far as 10 goes, 10 = 2*5, and 2 and 5 are both prime, so the prime factorization of 10 is 2*5. Now we replace the 8 and 10 with their prime factorizations in the fraction.

Next, we cancel out any common factors in the numerator and denominator.

Lastly, we multiply any leftover factors in the numerator together and any leftover factors in the denominator together.

We see that 8/10 reduced is 4/5, which we already knew because your friend told us so! However, now we know how they did it.

## Another Example

Let's look at another fraction that is a little more involved. Suppose we want to reduce the fraction 68/112. The first thing we want to do is find the prime factorization of the numerator and the denominator.

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