Question

Compare writing 0.3 as a fraction with writing $\mathbf{0}\mathbf{.}\overline{\mathbf{3}}$as a fraction.

Short answer

The fraction form of $\mathbf{0}\mathbf{.}\mathbf{3}$and $\mathbf{0}\mathbf{.}\overline{\mathbf{3}}\mathbf{}$are $\frac{\mathbf{3}}{\mathbf{10}}$and $\frac{\mathbf{1}}{\mathbf{3}}$respectively also fraction $\frac{\mathbf{1}}{\mathbf{3}}$is greater than the fraction$\frac{\mathbf{3}}{\mathbf{10}}$.

Step-by-step solution

Step 1. Given information

Considerthe number $0.3$and $0.\overline{3}$ as provided in the question

Step 2. Explanation

As provided number $0.3$is a terminating number and having 3 at tenth place

Therefore, the fraction form obtained as:

$\Rightarrow 0.3=\frac{3}{10}$

Also the provided number $0.\overline{3}$is a repeating number and repeat one digit 3

Let $x=0.333....$ equation $\left(1\right)$

Here one digit is repeating so multiply each side by $10$

$\Rightarrow 10x=3.333....$ equation $\left(2\right)$

subtract equation $\left(2\right)$from equation

$\left(1\right)$

$\begin{array}{l}\Rightarrow 10x-x=\mathrm{3.333....}-\mathrm{.333....}\\ \Rightarrow 9x=3\\ \Rightarrow x=\frac{3}{9}\\ \Rightarrow x=\frac{1}{3}\end{array}$

Also, $0.333...>0.300$

Therefore, fraction $\frac{1}{3}$is greater than the fraction $\frac{3}{10}$