Question

How do you convert 11.3 (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 answer:
Let 11.3 (3 being repeated) be ‘x’
So, $x=11.3$(3 being repeated) ……….(1)
Since there is only one recurring digit i.e. ‘3’
Hence multiplying ‘x’ by 10, we get
$10x=11.3\times 10=113.3$ ……….(2)
Subtracting equation (1) from equation (2) , we get
$10x-x=113.3-11.3$
(Subtracting the variables on the left side and the constants on the right side)
$\Rightarrow 9x=102$
By dividing both the sides by ’9’ we can get the fraction value of ‘x’ as
\[\Rightarrow \dfrac{9x}{9}=\dfrac{102}{9}\]
Cancelling out ‘9’ both from numerator and the denominator on the left side and dividing both the numerator and the denominator with the greatest common factor 3 on the right side, we get
$\begin{align}
  & \Rightarrow x=\dfrac{102\div 3}{9\div 3} \\
 & \Rightarrow x=\dfrac{34}{3} \\
\end{align}$
This is the required solution.

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 reducing the fraction \[\dfrac{102}{9}\], as we know the common factor of 102 and 9 is 3, so the greatest common factor of 102 and 9 is also 3. Hence dividing the numerator and the denominator with 3, we get $\dfrac{102}{9}=\dfrac{102\div 3}{9\div 3}=\dfrac{34}{3}$.