Question

How do you convert $0.13$ ($13$ repeating) to a fraction?

Answer 1

Hint: Here they have asked for converting a repeating number to a fraction, so we proceed by taking $0.13$ as any variable let’s say $x$. As $x$ is recurring or repeating in two decimal places, we multiply it by ${10^2} = 100$ for simplification purposes. Now, subtract the original equation which gives another equation, on simplifying this we get the required answer.

Complete step by step answer:
In this question, they have given that the number $13$ is repeating in $0.13$ which means we can write this repeating decimal number as $0.1313131313.....$ and now we need to find the value in fraction which gives such a recurring decimal.
Now, in order to solve this problem we consider as follows:
Let $x = 0.13$ ……….. equation (1)
Now look at the number and find how many digits are repeating or recurring after the decimal point and take that number of digits equal to $n$ and multiply the equation (1) by ${10^n}$.
So now we can see that in the given question we have two digits after the decimal that are repeating which is $13$, so we need to multiply the equation (1) with ${10^n}$ where $n = 2$ as two digits are repeating.
Therefore, we will get the multiplication factor as ${10^2} = 100$ and one thing we need to remember is that we have to keep the repeating numbers even after multiplication.
Therefore, multiplying the equation (1) with $100$, we get
$100x = 0.1313... \times 100$
$100x = 13.13$ ……….equation (2)
Now subtract the equation (1) from equation(2) which gives us the required fraction for the given decimal.
Therefore, $100x - x = 13.13 - 0.13$
$ \Rightarrow 99x = 13$
$ \Rightarrow x = \dfrac{{13}}{{99}}$

Hence the fraction $\dfrac{{13}}{{99}}$ gives us the decimal $0.13$ ($13$ repeating).

Note: Whenever we have this type of problem one thing we need to remember is that we need to keep the same repeating number after the decimal as we have done in the above. Because if we write just $13$ after the multiplication we don’t get the required answer, so this step is very important. After the subtraction of the equation, we need to get the integer value so we keep the decimal number even after multiplication.