Question

What is 0.31 (31 repeating) as a fraction?

Answer 1

Hint: Write the given repeating decimal as \[0.\overline{31}=0.313131......\]. Assume this expression as x = 0.313131…… and consider it as equation (1). Now, multiply both the sides with 100 and consider it as equation (2). Subtract equation (1) from equation (2) and divide both sides of the obtained difference with 99 to get the value of x in fractional form.

Complete step-by-step solution:
Here we have been provided with the decimal number 0.31 in which 31 is repeating, this means we have been provided with the decimal number \[0.\overline{31}\]. We are asked to write it in the fractional form.
Now, since 31 will repeat up to infinite places after the decimal point therefore we cannot directly remove the decimal. So, we need some other and better method. Let us assume the given decimal number as x. So, we have,
\[\Rightarrow x=0.\overline{31}\]
Removing the bar sign we have,
\[\Rightarrow x=0.313131......\] - (1)
Multiplying both the sides with 100 we get,
\[\Rightarrow 100x=31.313131.......\] - (2)
Subtracting equation (1) from equation (2) we get,
\[\begin{align}
  & \Rightarrow 99x=31.000..... \\
 & \Rightarrow 99x=31 \\
\end{align}\]
Dividing both sides with 99 we get,
\[\Rightarrow x=\dfrac{31}{99}\]
Hence, \[\dfrac{31}{99}\] represents the fractional form of the decimal number \[0.\overline{31}\].

Note: Note that the given number in the question is a rational number (non – terminating repeating) and that is why we were able to convert it in the fractional form. If the number is non – terminating and non – repeating then it is called an irrational number and we cannot write an irrational number into the fractional form. In the above question two digits were repeating just after the decimal point that is why we have multiplied the number with ${{10}^{2}}$. So in general, if n digits repeat after the decimal point then we need to multiply it with ${{10}^{n}}$.