Question

How do you convert the recurring decimal $ 0.\overline {32} $ to a fraction?

Answer 1

Hint:There is a repetition of 32 in the given number that is denoted by the bar on the digits 32, so the given number is irrational. In the given question we have to convert the number $ 0.\overline {32} $ to a fraction, this is done by multiplying the given number with a power of 10 such that the power is equal to the number of digits that are repeated and again multiply the obtained number with the same power of 10. By subtracting the obtained equations we can reach the correct answer.

Complete step by step answer:
The given number is $ 0.\overline {32} $ , it can be rewritten as $ 0.323232..... $ , let $ 0.\overline {32}
= R $
In the given question, the number of repeating digits is 2 (32). So we first multiply $ 0.\overline {32} $
with $ {10^2} $ and then the obtained number with $ {10^2} $ , then we subtract both the results with
each other, as follows –
 $
{10^2} \times R = {10^2} \times 0.323232.... \\
\Rightarrow 100R = 32.3232....\,\,\,\,\,\,\,\,\,...(1) \\
{10^2} \times (1) = {10^2} \times 32.3232... \\
\Rightarrow 10000R = 3232.3232.....\,\,\,\,\,\,...(2) \\
(2) - (1) \\
10000R - 100R = 3232.3232..... - 32.3232.... \\
\Rightarrow 9900R = 3200 \\
\Rightarrow R = \dfrac{{3200}}{{9900}} \\
\Rightarrow R = \dfrac{{32}}{{99}} \\
 $

Hence, the recurring decimal $ 0.\overline {32} $ is written in the fraction form as
 $ \dfrac{{32}}{{99}} $ .

Note: Rational numbers and Irrational numbers are the two types of real numbers. The numbers whose decimal expansion is terminating and non-repeating are called rational numbers and thus they can be expressed as a fraction such that the denominator is not zero, whereas irrational numbers are those which have repeating and non-terminating decimal expansion, the number $ 0.\overline {32} $ is irrational. So, the irrational numbers can be converted into a fraction using the method shown above.