Question

How do you write $ 0.41\overline{6} $ as a fraction? Also find the fraction form of $ 1.\overline{27},0.\overline{148} $ .

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 $ 0.41\overline{6} $ to a fraction. To understand the process better we also convert $ 1.\overline{27},0.\overline{148} $ to fractions.

Complete step by step answer:
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 $ 0.41\overline{6}=0.416666........ $
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 a decimal point and the non-recurring part of the number without a 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 the decimal and the number of digits of 0 is equal to the number of non-recurring digits in the given number after the decimal. The 9s come first and the zeroes come after that.
Now we find the fraction form of $ 0.41\overline{6} $ applying the rules.
 $ 0.41\overline{6}=\dfrac{416-41}{900}=\dfrac{375}{900}=\dfrac{5}{12} $ .
The fractional value of $ 0.41\overline{6} $ is $ \dfrac{5}{12} $ .
We now convert the decimals $ 1.\overline{27},0.\overline{148} $ .
 $ 1.\overline{27}=\dfrac{127-1}{99}=\dfrac{126}{99}=\dfrac{14}{11} $
\[0.\overline{148}=\dfrac{148-0}{999}=\dfrac{148}{999}=\dfrac{4}{27}\].

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 the decimal. That’s why we used two 9s and two 0s in the denominator.