Question

Convert the repeating decimal (recurring decimal) 0.3777777……. into a fraction.

Answer 1

In the question the decimal part of the number has the repeating (recurring) digit as 7 in it. So we have to slide the decimal point in this number to the right one place then we will have two recurring numbers which will help us in getting the fraction.

Complete solution step by step:
Firstly we form a simple equation by letting the number as
$N = 0.377777.....$ -----equation (1)

Now we have to multiply the number with ${10^t}$ where t is the number of digits which are not recurring or repeating in the initial number.

In our case we multiply the equation by 10 on both the sides only because there is only one number i.e. 3 which is not repeating. So we obtain an equation like this
$10N = 3.77777.....$ ----equation (2)

Looking at both the equations we can say that we are left with two recurring digit numbers.

Now what we do is, we try to eliminate the recurring part of the numbers. For that, we have to subtract the two equations in a way that left part of the equation is subtracted with left side of the other equation and right part of the equation is subtracted with the left side i.e.

Equation (2)-Equation(1)
$10N - N = 3.77777 - 0.377777 \\\ \Rightarrow 9N = 3.4 \\\ \Rightarrow N = \dfrac{{34}}{{90}} \\\$
So we have successfully converted the given recurring number into a fraction.

Note: A number whose digits are periodic and infinitely repeated number(s) are not zero is known as repeating or recurring decimal. It can be denoted by putting a dot or bar on the terminating digit(s) in this way-
0.33333... = 0.\mathop 3\limits^ \bullet \;{\text{or}}\;0.\mathop 3\limits^\\_
If a number’s decimal representation