Question

Write the decimal $0.3\overline{21}$as a fraction.

Short answer

The fraction number of $0.3\overline{21}$ is $\frac{53}{165}$.

Step-by-step solution

Step 1. Given information

The decimal number is $0.3\overline{21}$.

Step 2. Calculation 

Let x be $0.3\overline{21}$.

$x=0.3212121$ …… (1)

Clearly, 21 is the only digit which is repeating in equation (1)

Multiply equation (1) by 10 on both sides,

$10x=3.\overline{21}$

Consider, $3.\overline{21}=3+0.\overline{21}$ but $0.\overline{21}=\frac{21}{99}$.

$\begin{array}{c}3.\overline{21}=3+\frac{21}{99}\\ =\frac{\left(3\times 99\right)+21}{99}\\ =\frac{297+21}{99}\\ =\frac{318}{99}\end{array}$

Therefore,

$\begin{array}{c}10x=\frac{318}{99}\\ x=\frac{318}{990}\\ =\frac{53}{165}\end{array}$

Thus, the fraction number that can be formed is $\frac{53}{165}$.