Question

How do you convert $0.571428$($571428$ being repeated) to a fraction?

Answer 1

Hint: We will consider the fraction to be a variable $x$ then since there is repetition, we will multiply the fraction to remove the $6$ decimal places and then subtract the initial value from the original value of the fraction such that we can eliminate the recurring terms in the decimal and then we will simplify the expression to get the required solution.

Complete step by step solution:
We have the number given to us as:
$\Rightarrow 0.571428$ ($571428$ repeating)
We will consider the number to be $x$ therefore, it can be written as:
$\Rightarrow x=0.571428$ ($571428$ repeating)
Now to remove the recurring decimal place and since there are $6$ digits which are recurring, we will multiply both the sides of the equation by $100000$.
$\Rightarrow 100000x=571428.571428$ ($571428$ repeating)
It is to be kept in mind that even after multiplying, the number is still recurring.
We can see that the number will keep on recurring even after multiplication therefore, to remove the recurring decimals, we will subtract the original value $x$ from both sides of the equation.
$\Rightarrow 100000x-x=571428.571428-x$
On simplifying the left-hand side of the equation, we get:
$\Rightarrow 999999x=571428.571428-x$
On substituting the value of $x$ in the right-hand side of the equation, we get:
$\Rightarrow 999999x=571428.571428-0.571428$
On simplifying the right-hand side of the equation, we get:
$\Rightarrow 999999x=571428$
Note that in the above step we have removed the recurring decimal places. Now on transferring the term $999999$ from the left-hand side to the right-hand side, we get:
$\Rightarrow x=\dfrac{571428}{999999}$
Now the above term can be written as:
$\Rightarrow x=\dfrac{142857\times 4}{142857\times 7}$
On simplifying, we get:
$\Rightarrow x=\dfrac{4}{7}$, which is the required fraction.

Note: It is to be remembered that when a fraction is divided by the same number in the numerator and denominator, the value of the fraction does not change. The number which we have given in the question is a recurring number which is also called as an irrational number since its decimal never stops.