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 1, 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 1.
  • Although there is no right way which common divisor to use in the beginning, I would suggest using the first five (5) prime numbers in order as possible common divisor:

2, 3, 5, 7, 11, …

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 using Prime Factorization.

This is a simple illustration showing the fraction {8 \over {12}} 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.

4/8

Simplify using Method 1: Repeated Division Method

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

  • Divide the top and bottom by 2. 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 2 is {2 \over 4}. Can we stop here? Not yet! They can still be reduced by a second division of 2.

  • Divide again the top and bottom by 2. The answer is {1 \over 2} (as the simplest form of {4 \over 8} because the only divisor of its numerator and denominator is 1.
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 {4 \over 8} by dividing its numerator and denominator two times by the number 2. But wait! Is there a shortcut? Some of you may have observed that using a common divisor of 4 can directly simplify it with a single step!

In fact, the Greatest Common Factor (GCF) of this fraction is 4 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.

twelve over 18 or 12/18

Simplify using Method 1: Repeated Division Method

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

  • Divide the top and bottom numbers by the first prime number which is 2.
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 3. 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 = 6, 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.

90/150

Simplify using Method 1: Repeated Division Method

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

dividing both the numerator and denominator by 10, we get 90/150 = (90÷10)/(150÷10) = 9/15.

Now, 2 can’t divide both and so try 3.

we can then divide 9/15 by 3 to further simplify it. we can write this as 9/15 = (9÷3)/(15÷3) = 3/5.

Simplify using Method 2: Greatest Common Factor Method

Prime factorize each number and get the product of the common factors to obtain the needed GCF.

using prime factorization, we have 90 = 2×3×3×5 and 150 = 2×3×5×5.
next, we'll multiply the common factors of 90 and 150 which are 2, 3, and 5 together to find the GCF. therefore we have, GCF = 2×3×5 = 30.

Simplify the given fraction in one-step using the divisor GCF = 30.

90/150 = (90÷30)/(150÷30) = 3/5

Example 4: Simplify the fraction below.

6 over 21 or 6/21

Solution:

Divide the numerator and denominator by a common divisor of 3.

6/21 = (6÷3)/(21÷3) = 2/7

Example 5: Simplify the fraction .

18/81

Solution:

Simplify using repeated division method.

  • Divide both numerator and denominator by 3, two times!
18/81 = (18÷3)/(81÷3) = 6/27 = (6÷3)/(27÷3) = 2/9

Example 6: Simplify the fraction below.

70/126

Solution:

Simplify this fraction by the greatest common factor method.

  • Find the GCF by prime factoring both the numerator and denominator. Identify the common factors. Multiply them together to get the required GCF.
using prime factorization, we have 70 = 2×5×7 and 126 = 2×3×3×7.
we then multiply the common factors of 70 and 126 which are 2 and 7, to find the greatest common factor. we have GCF = 2×7 = 14.
  • After determining the GCF, divide the numerator and denominator to get the final answer.
70/126 = (70÷14)/(126÷14) = 5/9

Example 7: Simplify the fraction below.

84/105

Solution:

Find the greatest common factor between the numerator and denominator, and use this number to simplify the fraction.

  • Determine the GCF
we will use prime factorization to find the greatest common factor of 84 and 105. so we have 84 = 2×2×3×7 and 105 = 3×5×7.
the common factors are 3 and 7 which we will multiply together to find the GCF. therefore, GCF = 3×7 = 21.
  • Divide the numerator and denominator by GCF = 21.
84/105 = (84÷21)/(105÷21) = 4/5

Practice with Worksheets

You might also be interested in:

Adding and Subtracting Fractions with the Same Denominator
Add and Subtract Fractions with Different Denominators
Multiplying Fractions
Dividing Fractions
Equivalent Fractions
Reciprocal of a Fraction