Question

Express $0.12\overline{3}$ in its rational fraction form.

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 $0.12\overline{3}$ 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 answer:
The given decimal number is a representation of a non-terminating recurring decimal. 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 $0.12\overline{3}=0.1233333333........$
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 $0.12\overline{3}$ applying the rules.
$0.12\overline{3}=\dfrac{123-12}{900}=\dfrac{111}{900}=\dfrac{37}{300}$.

Therefore,the fractional value of $0.12\overline{3}$ is $\dfrac{37}{300}$.

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.