Question

How do you convert $7.\overline{12}$ to a fraction?

Answer 1

Hint: We first describe the concept of representation of non-terminating recurring decimals. We describe the process of converting them from decimal to fractions. Following those steps, we convert $7.\overline{12}$ to a fraction. To understand the process better we also take an example of another complicated form and change that decimal into fraction.

Complete step-by-step solution:
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 $7.\overline{12}=7.12121212........$
The process of converting into fraction form is below mentioned.
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 $7.\overline{12}$ applying the rules.
$7.\overline{12}=\dfrac{712-7}{99}=\dfrac{705}{99}=\dfrac{235}{33}$.
The fractional value of $7.\overline{12}$ is $\dfrac{235}{33}$.

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.