Question

Rewrite $0.\overline{5}$ as a simplified fraction.

• A. $\frac{4}{9}$
• B. $\frac{9}{5}$
• C. $\frac{5}{9}$
• D. $\frac{6}{9}$

Answer 1

The correct option is C $\frac{5}{9}$

Let, $x=0.\overline{5}$
$\Rightarrow x=\mathrm{0.555...}\cdots \left(i\right)$

Here, the repeating digit is 5.

Step 1: Place the repeating digit to the left of the decimal point by multiplying it by 10 because the unit digit (0.5) in the tenths place.
$\Rightarrow 10\times x=10\times \mathrm{0.555...}$
$\Rightarrow 10x=\mathrm{5.55...}\cdots \left(ii\right)$

Step 2: Place the repeating digit to the right of the decimal point.

In equation $\left(i\right),$ the repeating digit is already to the right.

Now, from $\left(ii\right)-\left(i\right),$ we get
$10x-x=\mathrm{5.555...}-\mathrm{0.555...}$
$\Rightarrow 9x=5$
Dividing both the sides by 9, we get
$\Rightarrow \frac{9x}{9}=\frac{5}{9}\Rightarrow x=\frac{5}{9}$