Equivalent Fractions

Equivalent fractions are different fractions that have different numerators and denominators but are equal to the same value. For example, 6/14 and 9/27 both are equivalent fractions as their decimal value (0.429) are the same.

The simplest form of all the equivalent fractions is the same. For example, the simplest form of 1/3, 2/6, and 3/9 is 1/3. Let's learn more about equivalent fractions, their examples, and others in detail.

What are Equivalent Fractions?

We define an equivalent fraction as the fraction that denotes the same value of any quantity but have different numerator and denominator are called an equivalent fraction.

Equivalent fractions indicate the same portion of the whole value. We can find an equivalent fraction for every fraction by multiplying or dividing both the numerator and the denominator with the same number. For instance, 3/9, 4/12, 5/15, and 6/18 are equivalent fractions as their values in the simplest form is 1/3.

Equivalent Fractions Examples

We represent the equivalent fraction as the fraction which results in the same value but it is interesting to note that they represent the same value when represented graphically.

We can understand with the help of the following example. Suppose we have to distribute a chapatti in two equal parts then each part is represented using a 1/2 fraction. Now, we distribute the similar chapatti into four parts a take two out of them which is represented as 2/4, similarly if the chapati is broken into six parts out of which three parts are taken which is represented as 3/6 and lastly if it is broken into eight parts and out of which four parts are taken then we represent this fraction as 4/8. Here in the above example, we see that all the fraction represents half of all the values, and thus all are equivalent fractions.

The image added below represents the equivalent fraction of 1/2.

Here, 1/2 = 2/4 = 3/6 = 4/8 all are equivalent fractions and their simplest form is 1/2.

How to Find Equivalent Fractions?

For every fraction, we can determine an equivalent fraction either by multiplying or dividing both the numerator and the denominator with the same number. That's why when all equivalent fractions are simplified, they are reduced to the same fraction. We have two ways of finding the equivalent fraction of any number,

• Multiplying the Numerator and Denominator by the Same Number
• Dividing the Numerator and Denominator by the Same Number

Multiplying the Numerator and Denominator by the Same Number

For every fraction, we can determine an equivalent fraction by multiplying both the numerator and the denominator with the same number. For instance, to determine the equivalent fraction of 2/3, multiply both the numerator and the denominator with the same number, i.e., 2. Thus, the equivalent fraction of 2/3 is 4/6. Similarly, we can find some other equivalent fractions by repeating the same process.

Equivalent fractions of 2/3

• Multiplying the Numerator and Denominator by 3, we get 2/3 × 3/3 = 6/9
• Multiplying the Numerator and Denominator by 4, we get 2/3 × 4/4 = 8/12
• Multiplying the Numerator and Denominator by 5, we get 2/3 × 5/5 = 10/15

Hence, we can conclude that equivalent fractions of 2/3 are 4/6, 8/12, and 10/15

Dividing the Numerator and Denominator by the the Same Number

For every fraction, we can determine an equivalent fraction by dividing both the numerator and the denominator by the same number. For instance, to find the equivalent fraction of 45/300, first, we have to find their common factors. Here, 3 is a common factor of both 45 and 300. So, the equivalent fraction of 45/300 can be found by dividing its numerator and denominator by 3. Thus, the equivalent fraction of  45/300 is 15/100. 

Let us simplify the fraction further.

5 is a common factor of 15 and 100. So, 15/100 = (15 ÷ 5)/(100 ÷ 5) = 3/20

Hence, the equivalent fractions of 45/300 are 15/100 and 3/20. As there are no common factors for 3 and 20 other than 1, 3/20 is the simplified form of 45/300.

Equivalent Fractions Chart

The chart that represents the equivalent fraction of any quantity is called the equivalent fraction chart. The chart added below represents the equivalent fraction of 1, 1/3, 1/6, etc.

From the given chart, we can observe that the equivalent fractions of 1/3 are 2/6, 4/12, 8/24,... Now, let us see the equivalent fractions of some unit fractions.

How to Check if Two Fractions Are Equivalent?

We can check whether two fractions are equivalent or not by using the three methods that include,

• Make Denominators of both Fraction Equal
• Determining the decimal form of both Fractions
• Cross Multiplication Method
• Visual Method

Making Denominators Equal

We can check whether the given fractions are equivalent by just making their denominator equal. And if they have an equal denominator along with an equal numerator they are equivalent fractions.

Example: Find if 4/10 and 6/15 are equivalent fractions or not.

Solution:

LCM of 10 and 15 is 30.

Multiply 4/10 by 3/3 and 6/15 by 2/2 to make their denominators equal to 30.

4/10 × 3/3 = 12/30
6/15 × 2/2 = 12/30

Now comparing two fractions we get that the denominator and the numerator of both the fractions are equal and hence, they are equivalent fraction.

Note: If the fractions are NOT equivalent, we can check the greater or smaller fraction by looking at the numerator of both the resultant fractions in which denominators are equal. Hence, this method can also be used for comparing fractions.

Determining the Decimal Form of Both Fractions

By finding the decimals of the given fractions, we can check whether they are equivalent or not. If the decimal form of both fractions is equal they are equivalent fractions and if the decimal form of both fractions is not equal they are not equivalent fractions.

Example: Find if 3/4, 6/8, and 12/16 are equivalent fractions or not.

Solution:

Decimal form of all the fractions is,

3/4 = 0.25
6/8 = 0.25
12/16 = 0.25

As the decimal values of the given fractions are the same, the given fractions are equivalent.

Cross Multiplication Method

To identify whether two fractions are equivalent or not, cross-multiplication is also used. The fractions are equivalent if both the products of the cross multiplication are the same.

Example: Find if 3/6 and 4/8 are equivalent fractions or not.

Solution:

The given fractions are equivalent as both the products of cross multiplication are 24.

Visual Method

By visualizing fractions we can easily find whether two fractions are equivalent or not as equivalent fractions represent the same shaded value in their visual structure. Such as 1/2 and 2/4 as shown in the image below both represent half of the value of any disc and thus they are equivalent fractions.

We simply observe the shaded portion of the given fractions and if they are equal our fraction is equivalent.

Difference Between Equivalent Fractions and Equal Fractions

Here’s a table that highlights the key differences between equivalent and equal fractions:

This comparison table should help clarify the distinctions between these two types of fractions,

Read More,

• Fractions
• Proper Fractions
• Improper Fraction
• Comparing Fractions

Example Problems on Equivalent Fractions

Example 1: Find "a" if 10/21 and a/42 are equivalent fractions.

Solution:

Given:
10/21 = a/42

Using Cross Multiplication

a × 21 = 10 × 42
a = 10 × 42 / 21
a = 20

Hence, the required value of "a" is 20

Example 2: Find the value of "x" if 5/36 and x/9 are equivalent fractions.

Solution: 

Given,
5/36 and x/9 are equivalent fraction

Using Cross Multiplication Method

5/36 = x/9
x = (5 × 9)/36
x = 5/4

Hence, the required value of "x" is 5/4

Example 3: Find if 5/9 and 11/15 are equivalent fractions or not.

Solution:

Let us use Cross Multiplication Method to check whether 5/9 and 11/15 are equivalent fractions or not.

Now, by Cross Multiplication of the given fractions we get,
5 × 15 = 75
9 × 11 = 99. 
Here, both products are not equal, i.e., 75 ≠ 99. 

Hence, 5/9 and 11/15 are not equivalent fractions.

Example 4: What are the equivalent fractions of 7/8?

Solution: 

To find equivalent fractions of 7/8, multiply the numerator and denominator by the same numbers.

7/8 × (2/2) = 14/16
7/8 × (3/3) = 21/24
7/8 × (4/4) = 28/32
7/8 × (5/5) = 35/40
7/8 × (6/6) = 42/48

Hence, the equivalent fractions of 7/8 are 14/16, 21/24, 28/32, 35/40, 42/48, and so on.

Example 5: What is its denominator, if the numerator of a fraction equivalent to 6/24 is 14?

Solution:

Let the unknown denominator be "x".

Given, 
6/24 = 14/x

We know that, if two fractions are equivalent their products when they are cross-multiplied are equal, i.e.
6 × x = 24 × 14
x = (24 × 14)/6 = 56.

Hence, the value of x is 56.

Practice Questions on Equivalent Fractions

Q1. Find an equivalent fraction for 3/4​ by multiplying the numerator and the denominator by 2.

Q2. Determine if 2/3​ and 8/12​ are equivalent fractions.

Q3. Simplify the fraction 18/24​ to its simplest form.

Q4. Find an equivalent fraction for 5/6 by multiplying the numerator and the denominator by 3.

Q5. Determine if 7/8​ and 21/24 are equivalent fractions.

Q6. Find two equivalent fractions for 4/5​.