Question

How do you convert $0.\overline{74}$ to a fraction?

Answer 1

Hint: The given decimal number is a representation of non-terminating recurring decimals. These types of decimal numbers are rational numbers. They can be expressed in the form of $\dfrac{p}{q}$. The expansion of the given decimal is $0.\overline{74}=0.7474747474........$ . We have to represent it in fractional form by following certain steps.

Complete step-by-step solution:
Let us now define the process of converting into fraction form using the steps as shown below.
Step: 1
We have to find the numerator part of the fraction where we take the difference between the whole number without decimal point and the non-recurring part of the number without decimal point.
Step: 2
We have to find the denominator part of the fraction where we take the digits of 9 and 0. The number of digits of 9 is equal to the number of recurring digits in the given number after decimal and the number of digits of 0 is equal to the number of non-recurring digits in the given number after decimal. The 9s come first and the zeroes come after that.
Now we find the fraction form of $0.\overline{74}$ applying the rules.
$0.\overline{74}=\dfrac{74-0}{99}=\dfrac{74}{99}$.
The fractional value of $0.\overline{74}$ is $\dfrac{74}{99}$.

Note: To understand the process better we take another example of $2.45\overline{74}$.
The fractional form of the decimal form will be $2.45\overline{74}=\dfrac{24574-245}{9900}=\dfrac{24329}{9900}$.
There are two recurring and two non-recurring digits in that number after decimal. That’s why we used two 9s and two 0s in the denominator.