Question

Add or subtract.

$0.\overline{03}-0.\overline{04}$.

Answer 1

Step 1: Find the fraction value of $0.\overline{03}$.

Given: $0.\overline{03}-0.\overline{04}$.

Let $\mathrm{x}=0.\overline{03}$

$\Rightarrow \mathrm{x}=0.030303...$ …(i)

Multiply by $100$ both sides

$\Rightarrow 100\times \mathrm{x}=100\times 0.030303...$

$\Rightarrow 100\mathrm{x}=3.030303...$ …(ii)

Subtract equation (i) from equation (ii)

$$\begin{gathered} \Rightarrow 100\mathrm{x}-\mathrm{x}=3.030303...-0.030303... \\ \Rightarrow 99\mathrm{x}=3 \\ \Rightarrow \mathrm{x}=\frac{3}{99} \end{gathered}$$

Hence, fraction form of $0.\overline{03}$ is $\frac{3}{99}$.

Step 2: Find the fraction value of $0.\overline{04}$.

Let $\mathrm{x}=0.\overline{04}$

$\Rightarrow \mathrm{x}=0.040404...$ …(i)

Multiply by $100$ both sides

$\Rightarrow 100\times \mathrm{x}=100\times 0.040404...$

$\Rightarrow 100\mathrm{x}=4.040404...$ …(ii)

Subtract equation (i) from equation (ii)

$$\begin{gathered} \Rightarrow 100\mathrm{x}-\mathrm{x}=4.040404...-0.040404... \\ \Rightarrow 99\mathrm{x}=4 \\ \Rightarrow \mathrm{x}=\frac{4}{99} \end{gathered}$$

Hence, fraction form of $0.\overline{04}$ is $\frac{4}{99}$.

Step 3: Find $0.\overline{03}-0.\overline{04}$.

Since, $0.\overline{03}=\frac{3}{99}$

$0.\overline{04}=\frac{4}{99}$

Now,

$$\begin{gathered} 0.\overline{03}-0.\overline{04} \\ =\frac{3}{99}-\frac{4}{99} \\ =\frac{3-4}{99} \\ =-\frac{1}{99} \end{gathered}$$

Hence, $0.\overline{03}-0.\overline{04}=-\frac{1}{99}$.