Question

How do you find the repeating decimal \[0.231\] with $31$ repeated as a fraction?

Answer 1

Hint: Consider the given number to be some constant (say x) and then multiply both sides by raising $10$ to the power of the number of digits repeated after decimal point (if two digits repeat then raise the power of $10\;{\text{to}}\;2$ ), then subtract the second equation from the original equation and then divide both sides with x coefficient, you will get the desired result.

Complete step by step solution:
 We have to take several steps to convert \[0.231\] with $31$ repeated to a fraction. Repeating or recurring decimals have their own way of being transformed into a fraction.
We have to assume the value of \[0.231\] with $31$ repeated in the first step to be $x$
$
   \Rightarrow x = 0.231\;(31\;{\text{being}}\;{\text{repeated}}) \\
   \Rightarrow x = 0.2313131...\; - - - - - - (i) \\
 $
Now we can see in the above equation that two digit i.e. $3\;{\text{and}}\;1$ are being repeated, so we will multiply the equation $10$ raise to the power of $2$ (Number of digits being repeated after decimal point).
So multiplying by ${10^2} = 100$ to both the sides,
$
   \Rightarrow 100 \times x = 100 \times 0.2313131... \\
   \Rightarrow 100x = 23.1313131...\; - - - - - - (ii) \\
 $
Now subtracting equation (i) from equation (ii), we will get
$
   \Rightarrow 100x - x = 23.1313131... - 0.2313131... \\
   \Rightarrow 99x = 22.9000000... \\
 $
Since only $0$ is repeating in the decimal, so we can remove $0$ and write $22.9000000... = 22.9$
$
   \Rightarrow 99x = 22.9 \\
   \Rightarrow 99x = 22.9 \times \dfrac{{10}}{{10}} \\
   \Rightarrow 99x = \dfrac{{229}}{{10}} \\
 $
Dividing both sides with coefficient of $x = 99$ to get the value of $x$
$
   \Rightarrow \dfrac{{99x}}{{99}} = \dfrac{{229}}{{99 \times 10}} \\
   \Rightarrow x = \dfrac{{229}}{{990}} \\
 $

Therefore the required fraction of the repeating number $0.2313131... = \dfrac{{229}}{{990}}$

Note: If more than one digit repeats after the decimal portion, it increases its power by the number of digits that are repeated after the decimal point in order to turn the decimal into a fraction number, when multiplying with the power of $10.$