# Equivalent Rational number

In this chapter we will learn about the concept of equivalent rational number & the method to identify them.

To understand the chapter, you should first have basic knowledge of rational number. Click the red link to understand the basics.

## What are equivalent rational numbers ?

The rational numbers are said to be equivalent when on simplification they have same values.

For example;
Consider the two rational number \mathtt{\ \frac{1}{2} \ \&\ \frac{3}{6}} .

Simplify both the fraction to decimal value.

\mathtt{\frac{1}{2}} ⟹ 0.5

\mathtt{\frac{3}{6}} ⟹ 0.5

You can see that both the above numbers have same decimal value. Hence they are equivalent rational numbers.

Conclusion
Rational numbers are equivalent when they have same values.

## Producing equivalent rational numbers

From a given rational number, you can generate multiple equivalent numbers by;

(a) Multiplying same number in numerator & denominator

(b) Dividing same number in numerator & denominator

Let us understand both the process with the help of example.

### Generating equivalent rational number by multiplication

Let \mathtt{\frac{a}{b}} is the given rational number.

Multiply numerator & denominator by integer p.

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{p\times a}{p\times b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{p.a}{p.b}}

Here the newly generated rational number \mathtt{\frac{p.a}{p.b}} and \mathtt{\frac{a}{b}} are the equivalent fraction.

I hope you understood the concept. Let us see some examples for better clarity.

Example 01
Find two equivalent rational number of \mathtt{\frac{7}{5}}

Solution
(i) First equivalent rational number.

Multiply numerator & denominator by 3.

\mathtt{\Longrightarrow \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 3}{5\times 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{21}{15}}

We got the fraction 21 / 15.

(ii) Second equivalent rational number

Multiply numerator and denominator by 4

\mathtt{\Longrightarrow \frac{7}{5}}\\\ \\ \mathtt{\Longrightarrow \ \frac{7\times 4}{5\times 4}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28}{20}}

We got the fraction 28 / 20.

Hence, \mathtt{\frac{7}{5} ,\ \frac{21}{15} \ \&\frac{28}{20}} are all equivalent rational number since on simplification they will produce same value.

Example 02
Fine two equivalent rational number of \mathtt{\frac{-3}{10}}

Solution

(a) First equivalent rational number

Multiply numerator and denominator by -3

\mathtt{\Longrightarrow \frac{-3}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times -3}{10\times -3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{9}{-30}}\\\ \\ \mathtt{\Longrightarrow \frac{-9}{30}}

Hence, -9/30 is the first equivalent rational number.

(b) Second equivalent rational number

Multiply numerator and denominator by 2.

\mathtt{\Longrightarrow \frac{-3}{10}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-3\times 2}{10\times 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{-6}{20}}

Hence, -6/20 is the second equivalent fraction.

Hence, \mathtt{\frac{-3}{10} ,\ \frac{-9}{30} \ \&\frac{-6}{20}} are all equivalent rational number since they have same values.

## Generating equivalent rational number by division

Let \mathtt{\frac{a}{b}} be the given rational number.

If we divide both numerator and denominator by integer m, we will get its equivalent number.

\mathtt{\Longrightarrow \frac{a}{b}}\\\ \\ \mathtt{\Longrightarrow \ \frac{a\div m}{b\div m}}

Let us solve some examples related to the concept.

Example 01
Find two equivalent rational number of \mathtt{\frac{6}{54}}

Solution
(a) First equivalent number.

Divide numerator & denominator by 3

\mathtt{\Longrightarrow \frac{6}{54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\div 3}{54\div 3}}\\\ \\ \mathtt{\Longrightarrow \ \frac{2}{18}}

Hence, 2/18 is the first equivalent rational number

(b) Second equivalent rational number

Divide numerator and denominator by 2

\mathtt{\Longrightarrow \frac{6}{54}}\\\ \\ \mathtt{\Longrightarrow \ \frac{6\div 2}{54\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{3}{27}}

So, 3/27 is the second equivalent number.

Hence, \mathtt{\frac{6}{54} ,\ \frac{2}{18} \ \&\ \frac{\ 3}{27}} are the equivalent rational numbers since all of them have same values.

Example 02
Find equivalent rational number of \mathtt{\frac{28}{14}}

Solution
(a) First equivalent fraction

Divide numerator & denominator by 7

\mathtt{\Longrightarrow \frac{28}{14}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28\div 7}{14\div 7}}\\\ \\ \mathtt{\Longrightarrow \ \frac{4}{2}}

Hence, we got one equivalent rational number 4 / 2.

(b) Second equivalent number

Divide numerator and denominator by 2.

\mathtt{\Longrightarrow \frac{28}{14}}\\\ \\ \mathtt{\Longrightarrow \ \frac{28\div 2}{14\div 2}}\\\ \\ \mathtt{\Longrightarrow \ \frac{14}{7}}

So, 14 / 7 is the second equivalent rational number.

Hence, \mathtt{\frac{28}{14} ,\ \frac{4}{2} \ \&\ \frac{\ 14}{7}} are the three equivalent rational number since they have same values.