Question

How do you convert $0.3$ ($3$ being repeated) to 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 answer:
We have to take several steps to convert $0.3$ ($3$ being 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 $0.3$ ($3$ being repeated) in the first step to be $x$
$x = 0.3\;(3\;{\text{being}}\;{\text{repeated}}) \\
\Rightarrow x = 0.3333333...\; - - - - - - (i) \\ $
Now we can see in the above equation that only one digit i.e. $3$ 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,
$10 \times x = 10 \times 0.3333333... \\
\Rightarrow 10x = 3.3333333...\; - - - - - - (ii)$
Now subtracting equation (i) from equation (ii), we will get
$10x - x = 3.3333333... - 0.3333333... \\
\Rightarrow 9x = 3.0000000... \\ $
Since only $0$ is repeating in the decimal, so we can remove $0$ and write $3.0000000... = 3$
$ \Rightarrow 9x = 3$
Dividing both sides with coefficient of $x = 9$ to get the value of $x$
$\dfrac{{9x}}{9} = \dfrac{3}{9} \\
\therefore x = \dfrac{1}{3} \\ $
Therefore the required fraction of repeating number $0.3333333... = \dfrac{1}{3}$.

Note:When more than one digit will repeat after the decimal part then in order to convert that decimal into fraction number, when multiplying with its power of $10$, raise its power by the number of digits that are being repeated after the decimal point.