Question

How do you convert $0.271$ ($71$ repeating) to a fraction?

Answer 1

Hint:
To convert the given number into a fraction, first we have to assume the repeating decimal number to be any variable. Then determine the repeating digit or digits in the number. Now, we have to place the repeating digits to the left of the decimal point. Then we have to shift the repeating digits to the left of the decimal point in the original number. We will get two equations, simplify the equation and find the value of the variable. The value obtained will be the final result.

Complete step by step solution:
We have to convert $0.271$ ($71$ repeating) to a fraction.
To do so, first we have to assume the repeating decimal number to be any variable.
So, Let $x = 0.271$…(i)
Now, determine the repeating digits in the number $0.271$.
Since, it is given that $71$ is repeated in the number $0.271$.
So, the repeating digits in the number $0.271$ is $71$. So, $x$ is recurring in $2$ decimal places.
Now, next we have to place the repeating digits ‘$17$’ to the left of decimal point in equation $x = 0.271$.
To do so, we have to shift the decimal point to the right of $27$. And this can be done by multiplying the equation $x = 0.271$ by $100$.
So, multiplying equation $x = 0.271$ by $100$, we get
$100x = 27.171$…(ii)
Now, we have to find the value of $x$.
We can find $x$ by subtracting equation (i) from (ii).
So, subtracting equation (i) from (ii) we get
$100x - x = 27.171 - 0.271$
Now, subtract both the left sides and right sides of the equation we get
$ \Rightarrow 99x = 26.9$
Now, we can get the value of $x$ by dividing both sides of the equation by $99$.
So, $x = \dfrac{{26.9}}{{99}}$
Thus, $0.2\overline {71} = \dfrac{{269}}{{990}}$.

Hence, $0.271$ ($71$ repeating) in fraction form is $\dfrac{{269}}{{990}}$.

Note:
In the above question, it should be noted that we shifted the decimal point to the right of $27$ instead of $2$. As questions state that $0.271$ is recurring in $2$ decimal places. So, we have to shift the decimal point to the right of $27$ by multiplying it by $100$.