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Simplifying 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 Fraction
Method 1: By Repeated Division Method 2: By Greatest Common Factor

* 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 to use the first five (5) prime numbers in order as possible common divisor:

list of the first five prime numbers are 2, 3, 5, 7, and 11

* 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 8/12 is being reduced to its simplest form. Can you see a pattern?

Original fraction Not yet! Simplest form
pie chart representation of the fraction 8/12 pie chart representation of the fraction 4/6 pie chart representation of the fraction 2/3

 

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


Example 1: Simplify the fraction 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.
divide both the numerator and denominator of 4/8 by 2

The output fraction after dividing the top and bottom by 2 is 2/4. 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 1/2 (as the simplest form of 4/8) because the only divisor of its numerator and denominator is 1. It's like the dead-end of possible common divisor.
divide numerator and denominator of 2/4 by 2 to simplify

 

Simplify using Method 2: Greatest Common Factor Method

In the above solution using repeated division, we have simplified 4/8 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.
divide top and bottom of 4/8 by 4 to get 1/2

 


Example 2: Simplify the fraction 12/18.

Simplify using Method 1: Repeated Division Method

Start simplifying using 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.
divide top and bottom of the fraction 12/18 by 2 to get 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.
divide numerator and denominator of 6/9 by 3 to obtain 2/3

 

Simplify using Method 2: Greatest Common Factor Method

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

 

finding the GCF of 12 and 18 using prime factorization method. The GCF of 12 and 18 is 6.

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

 

simplify 12/18 by dividing the numerator and denominator by 6 to get 2/3

 


Example 3: Simplify the fraction 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.

divide numerator and denominator of 90/150 by 10 to get 9/15, not simplified yet

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

simplify by the common divisor of 3 to arrive at 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.

 

the greatest common factor of 90 and 150 equals 30

 

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

divide the top and bottom of the fraction 90/150 by 30 to get the answer 3/5

 


Example 4: Simplify the fraction 6/21.

Solution:

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

 

common divisor is 3. simplest form of 6/21 = 2/7.

 

 


Example 5: Simplify the fraction 18/81.

Solution:

Simplify using repeated division method.

  • Divide both numerator and denominator by 3, two times!

 

do a repeated divisions of 3

 


Example 6: Simplify the fraction 70/126.

Solution:

Simplify this fraction by 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.

 

GCF of 70 and 126 is 14.
  • After determining the GCF, divide the numerator and denominator to get the final answer.

 

the fraction 70/126 is reduced to 5/9

 


Example 7: Simplify the fraction 84/105.

Solution:

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

  • Determine the GCF
The GCF of 84 and 105 is equal to 21.
  • Divide the numerator and denominator by GCF = 21.

 

84/105 is reduced to lowest term as 4/5.

 


 

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