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.