Question

How do you convert $$0.54$$ (54 repeating) to a fraction?

Answer 1

Hint: Given number here is a decimal number with two decimal places. Now we are asked to find the fraction form of this decimal number. But since the digits after decimal are repeating we can’t remove the decimal directly from the given number. For that we will just convert the number upto some extent to fraction form. Then to obtain the fraction form we need numbers that can be converted into fraction. So we will subtract the numbers given from its same form but having value greater than its own.

Complete step-by-step answer:
Given that $$0.54$$ is the number given with 54 repeating forms.
Repetition means we can write the number as $$0.54545454....$$
Let the number so given is $$x=0.\overline{5}\overline{4}$$
Then if we multiply both sides by 100 we get,
 $$100x=54.\overline{5}\overline{4}$$
Now we will find the difference between these,
 $$100x-x=54.\overline{5}\overline{4}-0.\overline{5}\overline{4}$$
Taking the difference,
 $$99x=54$$
Taking 99 on other side we get the fraction as,
 $$x={\displaystyle \frac{54}{99}}$$
Now simplifying this ratio, we will divide both the numbers by 9
 $$x={\displaystyle \frac{6}{11}}$$
Thus $$x=0.\overline{5}\overline{4}={\displaystyle \frac{6}{11}}$$
This is our final answer.
So, the correct answer is “${\displaystyle \frac{6}{11}}$”.

Note: Here note that we cannot find the fraction as we do it normally. That means just by multiplying the number by 100 or other power of 10, because here the numbers after decimal are in repetition pattern. That cannot be converted into fraction as $${\displaystyle \frac{54.\overline{5}\overline{4}}{100}}$$ . So we should solve this in the way mentioned above if the numbers are repeating.