 # Equivalent Fractions – Explanation & Examples

In mathematics, equivalent fractions are simply fractions with different numerators and denominators but represent the same proportion of a whole. Equivalent fractions seem to be different at a look, but they have similar or equal value.

For example, the equivalent fractions for 1/4 are:

2/8, 3/12, 4/16, etc.

The equivalent fractions have an equal amount or value after simplification of both their numerator and denominators. The fractions will generate the same value if cancellation by a common factor is made on both the numerator and denominator.

## What are Equivalent Fractions?

Equivalent fractions are two or more fractions that result in the same value after simplification. Suppose a/b and c/d are two fractions, then the fractions are equivalent only if the simplification of each fraction results in e/f.

In other words,

a/b = c/d = e/f.

For example, a fraction 1/3 has an equivalent of 5/15 because of simplification of 5/15 results in the same value as 1/3.

Now the question arises for why these fractions are equal despite having different numbers. The answer to this query is that the fractions contain numerators and denominators that are not co-prime numbers. Therefore they have a common multiple which on division produces the same value.

Let’s take an example:

1/2 = 2/4 = 4/8

You can notice that yet the above two factions have different integers, but after dividing both the numerator and denominator by a common factor, the result is:

(4 ÷ 4)/(8 ÷ 4)

=1/2

In this case, if we simplify 2/4, the result 1/2 again.

(2 ÷ 2)/(4 ÷ 2)

= 1/2

It has been shown that either dividing the denominator or multiplying the numerator with the same factor does not alter the value of the fraction. And therefore, equivalent fractions have an equal value when simplified.

## How do you find equivalent fractions?

Consider a case with the fraction 1/5.

Multiplying both the numerator and denominator with 2, 3 and 4 gives:

1/5 x 2/2 = 2/10

1/5 x 3/3 = 3/15

1/5 x 4/4 = 4/20

Therefore, it can be concluded that:

1/5 = 2/10 = 3/15 = 4/20

The equivalent fraction can only be generated by multiplication or division by a common factor. Carrying out addition or subtraction on the fraction only changes the value of a fraction.

Example 1:

Given that the fractions 5/16 and x/12 are equivalent calculate the value of x.

Solution

Given that:

5/16 = x/12

x = (5 x 12)/16

x = 60/16

x =15/4

And thus, the value of x is 15/4.

Example 2:

Find the value of x if the fractions 3/5 and 4/x are equivalent.

Solution

Given that,

3/5 = 4/x

x = (4 x 5)/3

x = 20/3

## Practice Questions

1. Write up to 5 equivalent fractions for each of the following:

(i) 3/4

(ii) 4/5

(iii) 6/7

(iv) 4/5

2. Find the equivalent fractions having a denominator of 12 for each of the following fractions.

(i) 1/2

(ii) 1/3

(iii) 3/4

(iv) 5/6

3. Change the following fractions into equivalent fractions having a value of 24 as their denominator:

(i) 6/12

(ii) 3/8

(iii) 2/6

(iv) 4/6

4. Identify the pairs of fractions that are equivalent and which are not:

(i) 2/3 and 8/12

(ii) 3/7 and 12/28

(iii) 5/8 and 15/27

(iv) 36/44 and 9/11

(v) 4/5 and 5/4

(vi) 5/8 and 27/18

5. I think of an equivalent fraction to 10/15 with 2 as the numerator. What fraction with a numerator of 2 am I thinking of?

6. Erick notices that either 3/5 or 3/4 is equal to the fraction 12/20. Which fraction is equal to 12/20?

7. James is giving his brother 2/5 of her nut collection. Calculate how many of 1/5 s of his nut collection is he giving his brother?

8. Peter gave 1/4 and 3/12 of orange to Donald and Pedro, respectively. Determine if he gave out an equivalent fraction of an orange.

9. John conducted a survey in his class and discovered that 56/96 of the students sampled out participated in sports after school. Express the fraction in its simplest form?

10. 7 is a prime number in a fraction 7/x. What number can replace x in this fraction so that it is not in simplest form?