Question

How do you convert $0.38$ ($8$ being repeated) to a fraction?

Answer 1

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

Complete step by step solution:
Repeating or recurring decimals have their own way of being converted into fraction, we have to follow some steps to convert $0.38$ ($8$ being repeated) to a fraction. In the first step, we have to assume the value of $0.38$ ($8$ being repeated) to be $x$
$\Rightarrow x = 0.38\;(8\;{\text{being}}\;{\text{repeated}}) \\\ \Rightarrow x = 0.3888888888...\; - - - - - - (i) \\\$

Now we can see in the above equation that only one digit i.e. $8$ is being repeated, so we will multiply the equation $10$ raise to the power of $1$ (Number of digits being repeated after decimal point).

So multiplying by ${10^1} = 10$ to both the sides,
$\Rightarrow x = 0.3888888888... \\\ \Rightarrow 10 \times x = 10 \times 0.3888888888... \\\ \Rightarrow 10x = 3.8888888888...\; - - - - - - (ii) \\\$

Now subtracting equation (i) from equation (ii), we will get
$\Rightarrow 10x - x = 3.8888888888... - 0.3888888888... \\\ \Rightarrow 9x = 3.5000000000... \\\$
Since only $0$ is repeating in the decimal, so we can remove $0$ and write $3.5000000000... = 3.5$
$\Rightarrow 9x = 3.5$

Dividing both the sides by coefficient of $x = 9$ to get the value of $x$
$\Rightarrow \dfrac{{9x}}{9} = \dfrac{{3.5}}{9} \\\ \Rightarrow x = \dfrac{{3.5}}{9} \\\$
Multiplying and dividing the right hand side by $10$ in order to get pure fraction
$\Rightarrow x = \dfrac{{3.5}}{9} \\\ \Rightarrow x = \dfrac{{3.5 \times 10}}{{9 \times 10}} \\\ \Rightarrow x = \dfrac{{35}}{{90}} \\\$

Simplifying it further,
$\Rightarrow x = \dfrac{{35}}{{90}} \\\ \Rightarrow x = \dfrac{{7 \times 5}}{{3 \times 3 \times 2 \times 5}} \\\ \Rightarrow x = \dfrac{7}{{18}} \\\$
Therefore the required fraction of repeating number $0.38888888... = \dfrac{7}{{18}}$

Note: we can write recurring or repeating numbers with help of bars as $0.38888888...$ can be written as $0.3\overline 8$. The bar should be given above the repeating digits if two digits are repeating $(e.g.\;0.232323232323....)$ then place the bar above both the repeating digits $(0.\overline {23} )$