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# How To Find Equivalent Fractions? (With Examples)

July 28, 2022

This post is also available in: हिन्दी (Hindi)

Equivalent fractions are the fractions that represent the same value, even though they look different. For example, if you have a cake, cut it into two equal pieces, and eat one of them, you will have eaten half the cake. If you cut a cake into eight equal pieces and eat four of them, you will still have eaten half the cake. These are equivalent fractions.

Let’s understand what is an equivalent fraction.

## What are Equivalent Fractions?

Equivalent fractions are the fractions that have different numerators and denominators but still, the values of the two fractions are equal.

For example, $\frac{1}{2}$ and $\frac {3}{6}$ are equivalent fractions. Also, $\frac{6}{12}$ is equivalent to both $\frac{1}{2}$ and $\frac {3}{6}$.  Both of these fractions $\frac {3}{6}$, $\frac{6}{12}$ can be reduced to the lowest form of $\frac{1}{2}$.

## Visual Representation of Equivalent Fractions

Let’s visualize fractions $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{3}{6}$ by dividing a circle into equal parts (sectors).

All these fractions have different numerators and denominators. But they all represent half part of a circle. Hence, the fractions $\frac{1}{2}$, $\frac{2}{4}$ and $\frac{3}{6}$ are equivalent fractions and they all reduce to the lowest/simplest form $\frac{1}{2}$.

Let’s consider one more example.

All these fractions also have different numerators and denominators. But they all represent three-fourths of a square. Hence, the fractions $\frac{3}{4}$, $\frac{12}{16}$ and $\frac{48}{64}$ are equivalent fractions and they all reduce to the lowest/simplest form $\frac{3}{4}$.

## How to Find Equivalent Fractions?

You can find equivalent fractions of any given fraction by

• multiplying both the numerator and the denominator by a number
• dividing both the numerator and the denominator by a number

Note:

• The method of multiplying is generally used when the numerator and denominator of the given fraction are smaller
• The method of dividing is generally used when the numerator and denominator of the given fraction are larger

### Finding Equivalent Fractions by Multiplying

In order to find the equivalent fractions of any fraction multiply both the numerator and the denominator by the same number.

Let’s understand it by the following example.

You want to find equivalent fractions of $\frac {5}{7}$. Let’s multiply both the numerator and the denominator by numbers $3$, $5$, $6$, $7$, $8$ and $10$.

Note: You can choose any number.

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $3$: $\frac {5 \times 3}{7 \times 3}$ = $\frac {15}{21}$

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $5$: $\frac {5 \times 5}{7 \times 5}$ = $\frac {25}{35}$

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $6$: $\frac {5 \times 6}{7 \times 6}$ = $\frac {30}{42}$

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $7$: $\frac {5 \times 7}{7 \times 7}$ = $\frac {35}{49}$

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $8$: $\frac {5 \times 8}{7 \times 8}$ = $\frac {40}{56}$

Multiplying the numerator and the denominator of $\frac {5}{7}$ by $10$: $\frac {5 \times 10}{7 \times 10}$ = $\frac {50}{70}$

$5$ equivalent fractions of $\frac {5}{7}$ are $\frac {15}{21}$, $\frac {25}{35}$, $\frac {30}{42}$, $\frac {35}{49}$, and $\frac {40}{56}$.

Note:

• For any fraction, there are countless equivalent fractions.
• You can find any number of equivalent fractions by multiplying the numerator and the denominator by the same number.
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### Finding Equivalent Fractions by Dividing

In order to find the equivalent fractions of any fraction divide both the numerator and the denominator by the same number.

Let’s understand it by the following example.

You want to find equivalent fractions of $\frac {42}{126}$.

Factors of $42$ are $1$, $2$, $3$, $6$, $7$, $14$, $21$, and $42$.

Factors of $126$ are $1$, $2$, $3$, $6$, $9$, $14$, $18$, $21$, $42$, $63$, and $126$.

Common factors of $42$ and $126$ are $1$, $2$, $3$, $6$, $14$, $21$, and $42$

You can divide numerator and denominator by any of these numbers – $2$, $3$, $6$, $14$, $21$, and $42$.

Note: $1$ is not considered as dividing by $1$ results in the same number.

Dividing the numerator and the denominator of $\frac {42}{126}$ by $2$: $\frac {42 \div 2}{126 \div 2}$ = $\frac {21}{63}$

Dividing the numerator and the denominator of $\frac {42}{126}$ by $3$: $\frac {42 \div 3}{126 \div 3}$ = $\frac {14}{42}$

Dividing the numerator and the denominator of $\frac {42}{126}$ by $6$: $\frac {42 \div 6}{126 \div 6}$ = $\frac {7}{21}$

Dividing the numerator and the denominator of $\frac {42}{126}$ by $14$: $\frac {42 \div 14}{126 \div 14}$ = $\frac {3}{9}$

Dividing the numerator and the denominator of $\frac {42}{126}$ by $21$: $\frac {42 \div 21}{126 \div 21}$ = $\frac {2}{6}$

Dividing the numerator and the denominator of $\frac {42}{126}$ by $42$: $\frac {42 \div 42}{126 \div 42}$ = $\frac {1}{3}$

$6$ equivalent fractions of $\frac {42}{126}$ are $\frac {21}{63}$, $\frac {14}{42}$, $\frac {7}{21}$, $\frac {3}{9}$, $\frac {2}{6}$ and $\frac {1}{3}$.

## How Do You Know Fractions Are Equivalent?

You can check whether two or more fractions are equivalent or not by simplifying the fractions. The idea behind the process of simplification is that every equivalent fraction reduces to the same fraction when written in the simplest form.

The methods that you can use to check whether given fractions are equivalent or not are

• Making Denominators Same
• Converting Fractions to Decimals
• Cross Multiplication Method

Let’s use these methods one by one.

### Making Denominators Same

The denominators of two or more equivalent fractions can be made the same by finding the L.C.M. of the denominators of the fractions.

Let’s consider the following examples to understand the process.

Ex 1: $\frac {12}{20}$ and $\frac {18}{30}$

The two denominators are $20$ and $30$.

L.C.M. of $20$ and $30$ is $60$.

For first fraction $60 \div 20 = 3$

For first fraction $60 \div 30 = 2$

Now, multiplying the numerators and the denominators of the first fraction by $3$ and the second by $2$.

$\frac {12 \times 3}{20 \times 3} = \frac {36}{60}$

$\frac {18 \times 2}{30 \times 2} = \frac {36}{60}$