How To Simplify Fractions

How To Simplify Fractions

Maths is a fun subject for some people, while scary for others. At cuemath, we endeavor to make math an easy subject by explaining the different concepts of Maths in an easy and simplified way. In this article, we will discuss how to simplify fractions.

What is a fraction?

A fraction is a number that represents a part of a whole. It is written in the form of two numbers separated by a fraction bar. To exemplify, a bowl of ice cream when shared by two friends, each will get 1/2 part. So, 1/2 is the fraction number representing the part of ice cream. The number written above the fraction bar is the numerator, and the number below the fraction bar is the denominator. A single fraction can be represented by different fractions when the numerator and denominator are changed in a similar way. Now let us learn how to simplify the fractions.

Types of Fractions

There are three types of fractions:

  1. Proper fractions: A fraction is a proper fraction when the number at the numerator is smaller than the number at the denominator position. For example, 3/5, 1/2, 5/9, 6/7, 11/13 are all proper fractions.
  2. Improper fractions: A fraction is called an improper fraction when the number at the numerator position is greater or equal to 1 the number at the denominator position. To illustrate, 5/2, 15/9, 50/7, 25/4, 18/5 are all examples of improper fractions.
  3. Mixed fractions: A fraction is called a mixed fraction, which is represented by a whole number and a fraction. As an example, 2 ½, 5 ½ are examples of mixed fractions.

What is a simplified fraction?

A simplified fraction in the form of a fraction that is reduced and in the lowest form and cannot be simplified further. In other words, a fraction is a simplified fraction when the numerator and denominator have a single common factor, which is 1. As an example, 1/3 is the simplified fraction of 2/6, 3/9, 4/12, 5/15, 7/21, etc. In the simplified fraction 1/3, there is only one common factor between 1 and 3, i.e., 1.

How to Simplify Fraction?

There are basically three methods that help to simplify the fraction. The three methods are:

  1. Repeated Division
  2. Greatest Common Factor (GCF)
  3. Prime Factor Tree

Method 1: The Repeated Division Method

In this method, the number is divided by a small common divisor. This is repeated a number of times that the common divisor that remains is 1. There is no specific rule to define which common divisor can get used to dividing the number. The first five prime numbers, i.e., 2, 3, 5,7, and 11 are usually used in increasing order.

Follow the below steps to simplify the fraction using the repeated division method:

Step1: Pick the smallest prime number.

Step2: Divide the numerator with the number chosen in Step1, a new fraction will get generated.

Step3: Again, divide the new fraction with the same number if it is divisible by the selected number. If not, then pick the other number to divide.

Step4: Repeat the process until there is no common factor in the fraction.

Examples of Repeated Division Method

Example 1: Simplify the fraction 4/8.

Suppose we have a fraction 4/8. Here both numerator and denominator are even numbers, so we can use 2 to divide the fraction.

So, 4/8 -> 4/2 (numerator divided by 2) and 8/2 (denominator divided by 2) -> 2/4. The new fraction we have is 2/4. We can divide this fraction by 2 again. Therefore, 2/4 -> 2/2 (numerator divided by 2) and 4/2 (denominator divided by 2) -> 1/2. The new fraction we get after division is 1/2.

1/2 cannot get simplified further as 2 is already a prime number and cannot be divisible further.

Example2: Simplify the fraction 15/25.

Here both numerator and denominator can be divided by 5. Therefore, we will divide the fraction by 5.

15/25 -> 15/5 (numerator divide by 5) and 25/5 (denominator divided by 5) -> 3/5. Now the fraction thus obtained 3/5 has a prime number as both numerators as well as the denominator. Therefore, it cannot get simplified further.

Hence, the simplified fraction of 15/25 is 3/5.

Method 2: Greatest Common Factor (GCF) Method

In this method, we divide the fraction by the greatest common factor and keep on repeating this until we reduce the fraction. As an example, in the example, above we divided 4/8 with 2 two times. But, we could have done it in one step by dividing the fraction by 4 because 4 is the greatest common factor between 4 and 8. You can follow the below steps to simplify the fraction:

Step 1: Make a list of factors of both numerator and denominator.

Step 2: Check the greatest common factor in the list of factors of numerator and denominator.

Step 3: Divide both numerator and denominator with the greatest common factor.

Step 4: It will give the simplified fraction.

Examples of Greatest Common Factor Method

Example1: Simplify the fraction 12/18.

First of all, find the factors of numerator and denominator.

Factors of 12 = 1,2,3,4,6,12

Factors of 18=1,2,3,6,9,18

The common factors of numerator and denominator are 1,2,3,6, out of which 6 is greatest. So, we have the greatest common factor as 6.

We will divide the fraction 12/18 with 6.

12/18 -> 12/6 (numerator divided by 6) and 18/6 (denominator divided by 6) -> 2/3

Hence, 2/3 is the simplified fraction of 12/18.

Example2: Simplify the fraction 25/75.

First of all, find the factors of numerator and denominator.

Factors of 25 = 1,5,25

Factors of 75=1,3,5,15,25,75

The common factors of numerator and denominator are 1,5,25, out of which 25 is greatest. So, we have the greatest common factor as 25.

We will divide the fraction 25/75 with 25.

25/75 -> 25/25 (numerator divided by 25) and 75/25 (denominator divided by 25) -> 1/3

Hence, 1/3 is the simplified fraction of 25/75.

Example 3: Simplify the fraction 8/10.

First of all, find the factors of numerator and denominator.

Factors of 8 = 1,2,4,8

Factors of 10=1,2,5,10

The common factors of numerator and denominator are 1,2 out of which 2 is greatest. So, we have the greatest common factor as 2.

We will divide the fraction 8/10 with 2.

8/10 -> 8/2 (numerator divided by 2) and 10/2 (denominator divided by 2) -> 4/5

Hence, 4/5 is the simplified fraction of 8/10.

Method 3: Prime Factor Tree Method

In the Prime Factor Tree method, we first find the prime factors of both numerator and denominator, and then cancel the common factors between them. Here, by the prime factor, we mean the prime number, which gets divisible by the number itself. In order to find the prime factor of any number, we will divide the number into numbers, of which one number needs to be a prime number. This process of division is repeated until we get both the numbers as prime numbers. Follow the steps below to simplify the fraction using the prime factor tree method.

Step 1: Find the prime factors of both numerator and denominator.

Step 2: Note down the prime factors of the numerator and denominator with multiply sign in between.

Step 3: Cancel the factors which are common to both numerator and denominator. The remaining number will be the simplified form of the fraction.

Examples of Prime Factor Tree Method

Example1: Simplify the fraction 24/60.

First of all, we will find the factors of 24 (numerator) such that one number is a prime number. So, 24 can be written as 2 x 12.

Next, we will find the factors of 12 in a similar way, and we will repeat the process until we get both numbers as prime numbers.

Factors of 12 = 2 x 6

Factors of 6 = 2 x 3

Now, 2 and 3 cannot be broken further. Therefore, factors of 24 are 2,2,2,3

Similarly, we will find the prime factors of denominator 60.

Factors of 60 = 2 x 30

Factors of 30 = 2 x 15

Factors of 15 = 3 x 5

So, prime factors of 60 are 2,2,3,5.

Next, we will write the prime factors as 2 x 2 x 2 x 3 / 2 x 2 x 3 x 5.

We will remove the common factors from the numerator and denominator. So, we will be left with a simplified fraction 2/5.

Example 2: Simplify the fraction 240/820.

Factors of 240 = 2 x 120 = 2 x 2 x 60 = 2 x 2 x 2 x 30 = 2 x 2 x 2 x 2 x 15 = 2 x 2 x 2 x 2 x 3 x 5

Factors of 820 = 2 x 410 = 2 x 2 x 205 = 2 x 2 x 5 x 41

Therefore, 2 x 2 x 2 x 2 x 3 x 5 / 2 x 2 x 5 x 41 = 2 x 2 x 3 / 41 = 12 /41

Conclusion

This was all about fractions and how to simplify fractions. These are very simple methods used to reduce the fractions, and you can use any method of these that you find easy to use. You will find many such interesting maths topics explained in a very easy way on Cuemath. So what are you waiting for, learn the math of numbers from cuemath in simplified form and make maths an interesting subject.