Question

How do you convert 0.013 (13 repeating) to a fraction?

Answer 1

Hint: Here in this question, we have to convert 0.013 to a fraction where 13 is on repetition. We need to form two equations for the conversion. In the first equation, we will let the given decimal be ‘x’. And the second equation needs to be formed by multiplying the whole equation with 100. After that both the equations will be subtracted and solved to get the answer.

Complete step by step answer:
Let’s discuss the question.
For this given decimal, we will form a linear equation in one variable in which we will find the value of that particular variable. We will be assigning the given number by ‘x’.
As 0.013 is a number which is repetitive and it is a 2 digit number. As there are 2 digits which are being repeated so we will multiply the whole equation by 100. If there were 3 digits in repetition, we would have multiplied the equation by 1000. Now, two equations will be formed in total. In the first equation, we will assign the given number by ‘x’. It will become our first equation and the second equation will be formed when we will multiply the first equation by 100 on both sides. In this equation, the decimal of 0.013 will be skipped right. After all these steps, we will subtract both the equations in order to find the value of ‘x’ which will be in the fraction. So, let’s start to solve this question now.
Let $x=0.013.....(i)$
Multiply the whole equation by 100. We will get:
$\Rightarrow 100\times x=100\times 0.013$
As we know that while multiplying with the 100, the decimal will shift to the right hand side up to 2 decimal places and 13 will be added at the end because of being a repetitive number. Now we will get:
$\Rightarrow 100x=1.313.......(ii)$
Next step is to subtract equation(i) from equation(ii). We will get:
$\Rightarrow 100x-x=1.313-0.013$
On solving further we will get:
$\Rightarrow 99x=1.3$
Now solve to find x:
$\Rightarrow x=\dfrac{1.3}{99}$
Now, remove the decimal and add zeros in the denominator:
$\Rightarrow x=\dfrac{13}{990}$
So this is our final answer.

Note: At final step, the fraction is already in lowest form. So there is no need to reduce anymore. Removing the decimal is basically addition of zeros in the denominator. In the second equation, we multiplied ‘x’ with 100 because 2 digits are in repetition. As the number of repeating digits will increase, the number of zeros also increases after 1 while multiplying with ‘x’.