Question

How do you find the repeating decimal 0.82 with 82 repeated, as a fraction?

Answer 1

Hint: Write the given repeating decimal as \[0.\overline{82}=0.828282....\]. Assume this expression as x = 0.828282……. and multiply both sides with 100. Now, subtract ‘x’ from ‘100x’ and divide both sides of the obtained difference with 99 to get the value of x in fractional form. This fractional form will be our answer.

Complete step by step answer:
Here, we have been provided with the decimal number 0.82 in which 82 is repeating, that means we have been provided with the decimal number \[0.\overline{82}\]. We are asked to write it in fractional form.
Now, sine 82 will repeat up to infinite places after the decimal therefore we cannot directly remove the decimal. So, we need some other and better approach. Let us assume the given decimal number as x. So, we have,
\[\Rightarrow x=0.\overline{82}\]
Removing the bar sign, we have,
\[\Rightarrow x=0.828282....\] - (1)
Multiplying both sides with 100 we get,
\[\Rightarrow 100x=82.828282....\] - (2)
Subtracting equation (1) from equation (2), we have,
\[\begin{align}
  & \Rightarrow 99x=82.0000.... \\
 & \Rightarrow 99x=82 \\
\end{align}\]
Dividing both sides with 99 we get,
\[\Rightarrow x=\dfrac{82}{99}\]
Hence, \[\dfrac{82}{99}\] represents the fractional form of the decimal number \[0.\overline{82}\].

Note:
One may note that this given number was a rational number and that is why we were able to convert it in the fractional form. You may see that here the digits were repeating after an interval of one digit that is why we multiplied both the sides with 100. If the digit would be repeating continuously like \[0.\overline{5}=0.555.....\] then we would have multiplied both the sides with 10. So, in general if the digits are repeated after \[{{n}^{th}}\] place then we will multiply both sides with \[{{10}^{n}}\].