# Repeating decimal to Fraction

in this post we will learn methods to convert repeating decimal into fraction.

## Convert repeating decimals into fractions

Let us review the basics of repeating decimals.

### What are repeating decimals?

The number whose decimal value repeats again and again are called repeating decimals.

These numbers are also called Recurring decimals.

### Examples of Repeating decimals

⟹ 7. 333333 . . .
It’s a repeating decimal since the decimal value 3 is repeating infinite times.
The number can be written as \mathtt{7.\overline{3}}

⟹ 8.75757575 . . .
It’s a repeating decimal since the value 75 is repeating again and again.
The number can be written as \mathtt{8.\overline{75}}

⟹ 65.13686868 . . .
It’s again a repeating decimal since the number 68 is repeating infinite times.
The decimal can be represented as: \mathtt{65.13\overline{68}}

### Methods to convert repeating decimal into fraction

Follow the below steps to convert recurring decimal into fraction.

Let the given recurring decimal is 2.3333 . . . .

(a) Write the decimal in the form of equation.
x = 2.3333 . . . .

(b) Identify the number of repeating digits.
In the given decimal 2.3333 . . ., there is only one repeating digit.

(c) Bring the repeating digit before the decimal point by multiplying with 10, 100, 1000 etc.

In the given decimal x = 2.3333. . . ., digit 3 is the repeating number. Multiply the equation with 10 to bring it before decimal point.

x = 2. 3333 . . .

Multiplying equation with 10.

10x = 23. 3333 . . . eq(1)

(d) Subtract the new equation with main equation.

On subtracting the two equations, we get;

9x = 21

x = 21/9

Hence the fraction value is 21/9.

Using the above method we have converted recurring decimal \mathtt{2.\overline{3}} into fraction 21/9.

I hope the process is clear, let us solve some question for conceptual clarity.

Example 01
Convert the recurring decimal \mathtt{18.\overline{7}} into fraction.

Solution
The decimal \mathtt{18.\overline{7}} signifies that digit 7 is repeated again and again.

The decimal can be written as;
⟹ 18.7777 . . . .

Follow the below step to convert it into fraction form;

(a) write the decimal into equation form.
x = 18.7777 . . . .

(b) Multiply by 10 to bring 7 before decimal point.

x = 18.7777 . . .

Multiplying 10 on both sides we get;
10x = 187.7777 . . . eq(1)

(c) Subtract new equation with main equation.

On subtracting the equation we get;

9x = 169

x = 169/9

Hence, the fraction value of decimal \mathtt{18.\overline{7}} is 169/9.

Example 02
Convert the decimal \mathtt{24.\overline{12}} into fraction.

Solution
The decimal \mathtt{24.\overline{12}} signifies that two digits are repeated again and again.

The decimal can be written as:
⟹ 24.12121212 . . . .

Follow the below steps to convert it into fraction.

(a) write the decimal into equation form.
x = 24.121212 . . . .

(b) Multiply by 100 to bring 12 before decimal point.

x = 24.121212 . . . .

Multiplying 100 on both sides;

100x = 2412.121212 . . .

(c) Subtract the new equation with main equation.

On subtracting the equation we get;

99x = 2388

x = 2388/99

Hence, the decimal is converted in fraction 2388/99.

Example 03
Convert the decimal \mathtt{52.\ 1\overline{45}}

Solution
The decimal can be written as;
⟹ 52.145454545 . . . .

(a) write the decimal into equation form
x = 52.1454545 . . . .

Multiply by 10 so that only recurring digits are present after decimal point.
10x = 521.454545 . . . -eq(1)

(b) Bring the repeating pair 45 in front of the decimal.
Multiply the equation by 1000.

1000x = 52145.454545 . . . . -eq(2)

(c) Subtract eq(2) with eq(1)

On subtracting the equation we get;

990x = 51624

x = 51624/990

Hence, the decimal is converted in fraction 51624/99

Example 04
Convert the recurring decimal \mathtt{73.\overline{641}} into fraction.

Solution
The decimal can be written as;
⟹ 73.641641641 . . .

(a) Write the decimal in form of equation.
x = 73.641641 . . .

(b) Bring the repeating digits 641 in front of decimal.
Multiply the equation by 1000.

x = 73.641641 . . .

Multiply 1000 on both sides of equation.

1000x = 73641.641641641 . . . -eq(1)

(c) Subtract eq(1) with main equation.

On subtracting the equations we get;

999x = 73568

x = 73568/999

Hence, the recurring decimal is represented as 73568/99.

Example 05
Convert the recurring decimal \mathtt{51.7\overline{32}} into fraction.

Solution
The decimal can be written as;
⟹ 51.732323232 . . . .

(a) Write the decimal in form of equation.
x = 51.7323232 . . .

Multiply the equation with 10 so that only repeating decimal remain on right of decimal point.
10x = 517.32323232 . . . – eq(1)

(b) Multiply the equation by 1000 to take one repeating pair 32 on left of decimal point.

x = 51.7323232 . . .

Multiply by 1000 on both sides of equation>

1000x = 51732.323232 . . . . – eq(2)

(c) Subtract eq(2) by eq(1)

On subtracting we get;

990x = 51215

x = 51215/90

Hence, 51215/990 is the required fraction.