Question

How do you convert 1.23 (3 being repeated) to a fraction?

Answer 1

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

Complete step by step answer:
Let 1.23 (3 being repeated) be ‘x’
So, $x=1.23$(3 being repeated) ……….(1)
Since there is only one recurring 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
$\begin{align}
  & \Rightarrow \dfrac{9x}{9}=\dfrac{11.1}{9} \\
 & \Rightarrow x=\dfrac{11.1}{9} \\
\end{align}$
Multiplying both numerator and denominator by 10 on the constant side, we get
$\begin{align}
  & \Rightarrow x=\dfrac{11.1\times 10}{9\times 10} \\
 & \Rightarrow x=\dfrac{111}{90} \\
\end{align}$
As both numerator and denominator are divisible by 3
So, it can be reduced as
$\begin{align}
  & \Rightarrow x=\dfrac{111\div 3}{90\div 3} \\
 & \Rightarrow x=\dfrac{37}{30} \\
\end{align}$

Note:
It should be remembered that the recurring decimal should be multiplied with the factor${{10}^{n}}$ where ‘n’ is the number of recurring digits. It should be subtracted from its initial value to remove the recurring 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.