How to Simplify Fractions

A fraction is considered to be “simplified” when it is expressed in the lowest term. That means the only common divisor between the numerator and denominator is [latex]1[/latex], and no other.

METHODS IN SIMPLIFYING FRACTIONS

Method 1: Simplify Fractions by Repeated Division

Keep dividing the numerator and denominator by a common divisor until such time that the only remaining common divisor is [latex]1[/latex].

Although there is no right way which common divisor to use in the beginning, I would suggest using the first five ([latex]5[/latex]) prime numbers in order as possible common divisor:

[latex]2[/latex], [latex]3[/latex], [latex]5[/latex], [latex]7[/latex], [latex]11[/latex], …

Method 2: Simplify Fractions Using the Greatest Common Factor

Find the greatest common factor (GCF) of the numerator and denominator.

Divide the top and bottom numbers of the fraction by the GCF to reduce to the lowest term.

You can find the GCF either by trial and error when the numbers are relatively small, or by using Prime Factorization.

This is a simple illustration showing the fraction [latex]\Large{8 \over {12}}[/latex] is being reduced to its simplest form. Can you see a pattern?

an illustration showing the fraction 8/12 reduced to its simplest form. from the original fraction, 8/12, the fraction can be reduced to 4/6 then down to its simplest form, 2/3. the pie charts in this image also visually illustrate that the shaded areas of the fractions 8/12, 4/6, and 2/3 are the same.

Let’s go over a few more examples with detailed explanations.

Examples of How to Simplify Fractions

Example 1: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

It is obvious that [latex]1[/latex] is not the only common divisor between the numerator and denominator. Since they are both even numbers, they must be divisible by [latex]2[/latex].

Divide the top and bottom by [latex]2[/latex]. Here’s what we got after doing so.

we can divide both the numerator and denominator by 2 to reduce the fraction 4/8 to its simplest form. therefore, 4/8 = (4÷2)/(8÷2) = 2/4.

The output fraction after dividing the top and bottom by [latex]2[/latex] is [latex]\Large{2 \over 4}[/latex]. Can we stop here? Not yet! They can still be reduced by a second division of [latex]2[/latex].

Divide again the top and bottom by [latex]2[/latex]. The answer is [latex]\Large{1 \over 2}[/latex] (as the simplest form of [latex]\Large{4 \over 8}[/latex] because the only divisor of its numerator and denominator is [latex]1[/latex].

since the fraction 2/4 can still be reduced, we will divide both the numerator and denominator by 2 to get its simplest form. thus we have, 2/4 = (2÷2)/(4÷2) = 1/2.

Simplify using Method 2: Greatest Common Factor Method

In the above solution using repeated division, we have simplified [latex]\Large{4 \over 8}[/latex] by dividing its numerator and denominator two times by the number [latex]2[/latex]. But wait! Is there a shortcut? Some of you may have observed that using a common divisor of [latex]4[/latex] can directly simplify it with a single step!

In fact, the Greatest Common Factor (GCF) of this fraction is [latex]4[/latex] because it is the LARGEST number that evenly divides the numerator and denominator. Because the numbers are small, the GCF can be determined by trial and error.

an easier way to simplify fractions is to divide both the numerator and denominator by its greatest common factor or GCF. for our fraction 4/8, the GCF of 4 and 8 is 4. therefore, we can divide both the numerator 4 and the denominator 8 by 4 t reduce it to its simplest form. we can write this as 4/8 = (4÷4)/(8÷4) = 1/2.

Example 2: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

Start simplifying using the first few prime numbers ([latex]2[/latex], [latex]3[/latex], [latex]5[/latex], [latex]7[/latex], [latex]11[/latex], etc).

Divide the top and bottom numbers by the first prime number which is [latex]2[/latex].

from our original fraction 12/18, we will divide both the numerator and denominator by 2 to reduce it to its simplest form. so we have, 12/18 = (12÷2)/(18÷2) = 6/9.

We still have a common divisor! Divide the top and bottom by the next larger prime number which is [latex]3[/latex]. We should get the final answer after this step.

we can still further reduce the fraction 6/9 using their common divisor which is 3. dividing both the numerator and denominator by 3, we get 6/9 = (6÷3)/(9÷3) = 2/3.

Simplify using Method 2: Greatest Common Factor Method

To find the greatest common divisor, we are going to perform prime factorization on each number. Next, identify the common factors between them. Finally, multiply the common factors to get the required GCF that can simplify the fraction.

using prime factorization, we have 12 = 2×2×3 and 18 = 2×3×3.

we then multiply their common factors which are 2 and 3 to find the greatest common factor or GCF. so we get, GCF = 2×3 = 6.

Since GCF = [latex]6[/latex], use this number to divide the numerator and denominator to get the answer in a single step.

we will use our GCF which is 6 to divide the numerator and denominator of our original fraction, 12/18, to easily reduce it to its simplest form. therefore we have, 12/18 = (12÷6)/(18÷6) = 2/3.

Example 3: Simplify the fraction below.

Simplify using Method 1: Repeated Division Method

We can start testing numbers [latex]2[/latex], [latex]3[/latex], [latex]5[/latex], etc. to simplify this. But there is an obvious divisor that stands out! Since both numbers end with zero, they should be divisible by [latex]10[/latex].