Question

I am wondering how I can change a repeating decimal to a fraction ?

Answer 1

Hint: Take the repeating decimal as ‘x’ and mark it as equation (1). Multiply the repeating decimal by ${10}^n$ where ‘n’ is the number of repeating digits and mark it as equation (2). Subtract both the equation to eliminate the repeating part. Solve for ‘x’ to convert to fraction by doing the necessary simplifications.

Complete step-by-step answer:
It can be explained by taking a suitable example.
Let 1.23 ( where ‘3’ being repeated) be ‘x’
So, $x=1.23$(3 being repeated) ……….(1)
Since there is only one repeating digit i.e. ‘3’
Hence multiplying ‘x’ by 10, we get
$10x=1.23\times 10=12.33$ ……….(2)
Subtracting equation (1) from equation (2) , we get
$10x-x=12.33-1.23$
(Subtracting the variables on the left side and the constants on the right side)
$\Rightarrow 9x=11.1$
By dividing both the sides by ’9’ we can get the fraction value of ‘x’ as
$\begin{array}{rl}& \Rightarrow {\displaystyle \frac{9x}{9}}={\displaystyle \frac{11.1}{9}}\\ & \Rightarrow x={\displaystyle \frac{11.1}{9}}\end{array}$
Multiplying both numerator and denominator by 10 on the constant side, we get
$\begin{array}{rl}& \Rightarrow x={\displaystyle \frac{11.1\times 10}{9\times 10}}\\ & \Rightarrow x={\displaystyle \frac{111}{90}}\end{array}$
As both numerator and denominator are divisible by 3
So, it can be reduced as
$\begin{array}{rl}& \Rightarrow x={\displaystyle \frac{111÷3}{90÷3}}\\ & \Rightarrow x={\displaystyle \frac{37}{30}}\end{array}$

Note: It should be remembered that the recurring decimal should be multiplied with the factor${10}^n$ where ‘n’ is the number of repeating digits. It should be subtracted from its initial value to remove the repeating part. For converting the fraction to decimal first the numerator and denominator should be multiplied by ${10}^m$ where ‘m’ is the number of digits after the decimal point. Then reduce the fraction , if necessary.