Question

Add or subtract.

$-0.\overline{63}+\frac{5}{99}$.

Answer 1

Step 1: Find the fraction form of $-0.\overline{63}$.

Given: $-0.\overline{63}+\frac{5}{99}$.

Let $\mathrm{x}=-0.\overline{63}$

$\Rightarrow \mathrm{x}=-0.636363...$ …(i)

Multiply by $100$ both sides

$\Rightarrow 100\times \mathrm{x}=100\times \left(-0.636363...\right)$

$\Rightarrow 100\mathrm{x}=-63.636363...$ …(ii)

Subtract equation (i) from equation (ii)

$$\begin{gathered} \Rightarrow 100\mathrm{x}-\mathrm{x}=\left(-63.636363...\right)-\left(-0.636363...\right) \\ \Rightarrow 100\mathrm{x}-\mathrm{x}=-63.636363...+0.636363... \\ \Rightarrow 99\mathrm{x}=-63 \\ \Rightarrow \mathrm{x}=-\frac{63}{99} \end{gathered}$$

Hence, fraction form of $-0.\overline{63}$ is $-\frac{63}{99}$.

Step 2: Find $-0.\overline{63}+\frac{5}{99}$.

Since, $-0.\overline{63}=-\frac{63}{99}$

Now,

$$\begin{gathered} -0.\overline{63}+\frac{5}{99} \\ =-\frac{63}{99}+\frac{5}{99} \\ =\frac{-63+5}{99} \\ =-\frac{58}{99} \end{gathered}$$

Hence, $-0.\overline{63}+\frac{5}{99}=-\frac{58}{99}$.