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 . . .

Follow the below steps;

**(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**.