# Reducing Fractions Through Factorization

Reducing fractions is another way of saying, “finding the **lowest equivalent fraction**.” That’s because you are normally trying to reduce a **proper fraction** to its simplest term.

**Note**: The term “**reduce**” usually refers to proper fractions (the numerator is **smaller** than the denominator). Whereas, the term “**simplify**” generally refers to improper fractions (the numerator is **larger** than the denominator).

When reducing a proper fraction, follow these steps:

- Factor the numerator.
- Factor the denominator.
- Cancel-out
**fraction mixes**that have a value of 1. - Re-write your answer as the reduced fraction.

**Let’s use an easy example of reducing fractions so you will get the idea…**

Notice that the original answer to adding the fractions our sample problem is “2/4.” To determine if our answer is in its simplest form, we must factor the numerator and the denominator into its prime numbers.

Click here for a review of prime numbers.

The factors of a number are numbers that when multiplied together will equal that number. The easiest way to be sure that you have accounted for ALL of the factors of a number contained in a fraction is to break them down into prime factors.

What we are** looking for** are the prime numbers that are common factors in both the numerator and the denominator of a fraction. If we find these common factors, we can then cancel them out. The results will be the lowest fractional equivalent fraction.

Since “2” is a common factor in both the numerator and denominator of our example, it indicates that our answer is not a fraction in its simplest form. Therefore, we will cancel out (/) one of the 2’s in both the numerator and denominator by dividing by “2”. The results is a reduced fraction in its simplest form.

**Here’s the Rule…**

**Here’s Another Example:**

To reduce **24/56** we factor the Numerator (**24** = 2 x 2 x 2 x 3) and then factor the Denominator (**56** = 2 x 2 x 2 x 7).

In this example all of the **2s** are eliminated because there are an equal number of 2s in both the numerator and denominator. They **cancel-out one-for-one**. That’s what we mean by a **fraction mix** that has the **value of “1”**.

The **correct answer** for the example above is a reduced fraction that’s equal to **3/7**.

**24/56**.

**You already know that 2/2 = 1, so…**

is the same as

which is equal to** 1 x 1 x 1 x 3/7**

Therefore, you would re-write your answer as **24/56** is equal (or equivalent) to **3/7**.

That’s the **basics** of reducing proper fractions, regardless of size.

### Always keep in mind…

Whatever you do to the numerator of a fraction you must also do to the fraction’s denominator. So if you have to divide the numerator by a number, you must also divide the denominator by the same number. That way you will not change the overall value of the fraction.

**Let’s add a little tougher fraction to be sure you’ve got it…**

In this problem, a “2” and a “3” can be found as factors in both the numerator and the denominator of a fraction. Notice how we only cancel-out one-for-one! First we divide the numerator and denominator by “2”, then divide both the numerator and denominator by “3.” So, what’s left in the numerator i**s 1 x 1 x 3 = 3** and the denominator is **1 x 2 x 2 x 1 = 4**. That leaves use with a reduced fraction equal to **3/4**.

**A sure-fire check that you fraction is in lowest terms is when the Greatest Common Factor (GCF) of its numerator and denominator is 1.**