Question

$1.21$($21$ repeating) as a fraction?

Answer 1

Expressing $1.21$($21$ repeating) as a fraction.

First, write ($21$ repeating) as $1.\overline{21}$.

Let $x=1.\overline{21}$

Next, multiply each side by $100$ giving:

$100x=121.\overline{21}$

Then subtract each side of the first equation from each side of the second equation giving:

$$\begin{gathered} 100\mathrm{x}-\mathrm{x}=121.\overline{21}-1.\overline{21} \\ \Rightarrow (100-1)\mathrm{x}=120 \\ \Rightarrow 99\mathrm{x}=120[\mathrm{Divide}\mathrm{both}\mathrm{sides}\mathrm{by}99] \\ \Rightarrow \frac{99\mathrm{x}}{99}=\frac{120}{99} \\ \Rightarrow \mathrm{x}=\frac{3\times 40}{3\times 33}[\mathrm{Cancel}\mathrm{out}\mathrm{the}\mathrm{common}\mathrm{term}] \\ \therefore \mathrm{x}=\frac{40}{33} \end{gathered}$$

Hence, $1.21$($21$ repeating) as a fraction is $\frac{40}{33}$.