A given fraction and the fraction obtained by multiplying (or dividing) both its numerator and denominator by the same non-zero number are called equivalent fractions.

$\frac{1}{2}$ , $\frac{2}{4}$ , $\frac{3}{6}$ are examples of equivalent fractions.

Equivalent fractions are found by multiplying both the numerator and the denominator by the same number. Equivalent fractions can also be found by dividing both the numerator and the denominator by the same number.

## What is an Equivalent Fraction?

Look at the representation of a few fractions:

The shaded areas in these figures represent the fractions, $\frac{1}{2}$ , $\frac{2}{4}$ , $\frac{3}{6}$ respectively. If we place one diagram over the other, they are found to be equal. That is, the fractions representing these shaded areas are equal. Such fractions are called as equivalent fractions.

If two fractions are equivalent, then the product of the numerator of the first and the denominator of the second is equal to the product of the numerator of second and the denominator of the first fraction.

In other words, if $\frac{a}{b}$ = $\frac{c}{d}$ , then we can say that ad = bc.

## How to Find Equivalent Fractions

We can see that the fraction, $\frac{3}{6}$ = $\frac{1\times3}{2\times3}$ . Here, we multiplied the numerator and the denominator of the fraction $\frac{1}{2}$ by 3, and we multiplied the numerator and denominator of $\frac{1}{2}$ by 2 to get $\frac{2}{4}$.

In other words, if we divide the numerator and denominator of $\frac{3}{6}$ by 3, we get; $\frac{3}{6}$ = $\frac{\frac{3}{3}}{\frac{6}{3}}$ = $\frac{1}{2}$

We can find the equivalent fractions of a given fraction in two different ways:
• To find an equivalent fraction of a given fraction, we have to multiply both the numerator and the denominator of the given fraction by the same number.
• To find an equivalent fraction, we may divide both the numerator and the denominator by the same number.

## Equivalent Fractions Examples

Given below are some solved problems on equivalent fractions.

Example 1:

Find three equivalent fractions for $\frac{2}{3}$

Solution:
To find the equivalent fractions for $\frac{2}{3}$ , we shall multiply the numerator and denominator by the same number.
$\frac{2}{3}$ = $\frac{2\times2}{3\times2}$ = $\frac{4}{6}$

$\frac{2}{3}$ = $\frac{2\times3}{3\times3}$ = $\frac{6}{9}$

$\frac{2}{3}$ = $\frac{2\times10}{3\times10}$ = $\frac{20}{30}$

Example 2:

Find the equivalent fraction of $\frac{5}{35}$ with the denominator 7.

Solution:
Here the denominator of the given fraction is 35. We can divide 35 by 5 to get 7. So, to find the equivalent fraction of $\frac{5}{35}$ , we have to divide both numerator and denominator by 5

$\frac{5}{35}$ = $\frac{\frac{5}{5}}{\frac{35}{5}}$

= $\frac{1}{7}$

## Generation of Equivalent Fraction and Reasoning

There are occasions when mathematical domains could be related to one another. The solution of missing value problem ($\frac{m}{n}$: $\frac{p}{x}$), equivalent fraction generation and the generation of the second ratio which reflects an equal probability or value would be having much in common.

Likewise, the unit rate in a proportional situation could be related examples like the slope(m) in a linear function with an equation of the form y = mx.

### Points to remember while generating equivalent fractions:

• Equivalent fractions have the same value.
• Any fraction would have many equivalent fractions.
• Some equivalences are more useful than others.

While forming or generating the equivalences the following things are necessary to watch out for:

• Recognising the fractions to be made into equivalence
• Forming and recording simple equivalent fractions.

### Examples for generating equivalent fractions:

Given below are some examples for generating equivalent fractions.

Example 1:

Complete $\frac{6}{7}$ : $\frac{x}{28}$

Solution:

7 in the denominator must be increased four times to give 28.
The numerator must be treated similarly and increased four times.

$\frac{6}{7}$ = $\frac{6\times4}{7\times4}$

$\frac{24}{28}$

X = 24

Example 2:

Convert 16 $\frac{3}{5}$ to an improper fraction.

Solution:

16 $\frac{3}{5}$ = 16 + $\frac{3}{5}$ = 13 × ($\frac{5}{5}$) + $\frac{2}{5}$

($\frac{65}{5}$) + ($\frac{2}{5}$)

$\frac{67}{5}$

## Recognition and Reasoning of Equivalent Fractions

Fractions that have the same value though they look different are called equivalent fractions. Equivalent fractions have same value, because when we multiply or divide both the numerator and denominator by the same number, the fractions have the same value only.

Points to remember:
• The Numerator and the denominator must contain a whole number.
• To check whether the fractions are equivalent or not, we need to multiply or divide the numerator and denominator by the same number.
• Do not add or subtract numbers from the numerator and the denominator to make the fraction equivalent.

### Examples on Recognizing Equivalent Fractions

Given below are some of the examples for recognition and reasoning of equivalent fractions:

Example 1:

Find the missing number that makes the fraction equivalent?
$\frac{1}{3}$ = $\frac{5}{?}$

Solution:

Step 1: Since the given fractions are equivalent, we can multiply and divide both the numerator and denominator by the same number.

Step 2: As 1(first fraction numerator) times 5 $\rightarrow$ 5(second fraction numerator)
So, multiply both the numerator and the denominator of the first fraction by 5.

Step 3: $\frac{1 \times 5}{3 \times 5}$ = $\frac{5}{15}$

Step 4: So, the missing number = 15.

Example 2:

Find the missing number that makes the fraction equivalent?
$\frac{?}{9}$ = $\frac{2}{3}$

Solution:

Step 1: Since the given fractions are equivalent, we can multiply and divide both the numerator and denominator by same number.

Step 2: As 3(second fraction denominator) times 3 $\rightarrow$ 9(first fraction denominator)
So, multiply both the numerator and the denominator of the second fraction by 3.

Step 3: $\frac{2 \times 3}{3 \times 3}$ = $\frac{6}{9}$

Step 4: The missing number = 6.

Example 3:

Which of the following fraction is equivalent to $\frac{4}{5}$?
a) $\frac{6}{8}$

b) $\frac{5}{6}$

c) $\frac{8}{10}$

d) $\frac{2}{3}$

Solution:

Step 1: Given fraction: $\frac{4}{5}$

Multiply the numerator and denominator by 2

Step 2: $\frac{4 \times 2}{5 \times 2}$ = $\frac{8}{10}$

Step 3: Option C is the correct answer.

Example 4:

Find the missing number to complete the pattern of equivalent fractions.
$\frac{1}{3} = \frac{2}{6} = \frac{3}{?} = \frac{4}{12}$

Solution:

Step 1: Let’s equate first fraction and the third fraction $\frac{1}{3} = \frac{3}{?}$

Step 2: As 1(first fraction numerator) times 3 $\rightarrow$ 3(third fraction numerator)
So, multiply both the numerator and the denominator of the first fraction by 3

Step 3: $\frac{1 \times 3}{3 \times 3}$ = $\frac{3}{9}$

Step 4: So, the missing number = 9.

Example 5:

Find the missing number to complete the pattern of equivalent fractions.
$\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{?}$

Solution:

Step 1: Let’s equate first fraction and the fourth fraction

$\frac{1}{2} = \frac{4}{?}$

Step 2: As 1(first fraction numerator) times 4 => 4(fourth fraction numerator)

So, multiply both the numerator and the denominator of the first fraction by 4.

Step 3: $\frac{1 \times 4}{2 \times 4}$ = $\frac{4}{8}$

Step 4: So, the missing number = 8.

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