Question

Show $1.\overline{21}$ ($21$ repeating) as a fraction.

Answer 1

Solve for the required value:

Let $x=1.\overline{21}...\left(1\right)$

Multiplying by $100$ we get,

$100x=121.\overline{21}...\left(2\right)$

Subtracting $\left(1\right)$ from $\left(2\right)$ we get,

$$\begin{gathered} 100x-x=121.\overline{21}-1.\overline{21} \\ \Rightarrow 99x=120 \\ \Rightarrow x=\frac{120}{99} \\ \Rightarrow x=\frac{40}{33} \end{gathered}$$

Hence, $1.\overline{21}$ can be written as $\frac{40}{33}$ in fraction.