Question

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

Answer 1

Hint: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 $x$
$$\begin{gathered} \Rightarrow x=0.38(8\text{being}\text{repeated}) \\ \Rightarrow x=0.3888888888...------(i) \end{gathered}$$

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,
$$\begin{gathered} \Rightarrow x=0.3888888888... \\ \Rightarrow 10\times x=10\times 0.3888888888... \\ \Rightarrow 10x=3.8888888888...------(ii) \end{gathered}$$

Now subtracting equation (i) from equation (ii), we will get
$$\begin{gathered} \Rightarrow 10x-x=3.8888888888...-0.3888888888... \\ \Rightarrow 9x=3.5000000000... \end{gathered}$$
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 $x$
$$\begin{gathered} \Rightarrow {\displaystyle \frac{9x}{9}}={\displaystyle \frac{3.5}{9}} \\ \Rightarrow x={\displaystyle \frac{3.5}{9}} \end{gathered}$$
Multiplying and dividing the right hand side by $10$ in order to get pure fraction
$$\begin{gathered} \Rightarrow x={\displaystyle \frac{3.5}{9}} \\ \Rightarrow x={\displaystyle \frac{3.5\times 10}{9\times 10}} \\ \Rightarrow x={\displaystyle \frac{35}{90}} \end{gathered}$$

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

Note: we can write recurring or repeating numbers with help of bars as $0.38888888...$ can be written as $0.3\bar{8}$. 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 $(0.\overline{23})$