Question

How do you convert $ 0.84 $ (84 being repeated to a fraction?

Answer 1

Hint: In order to determine the fraction of a repeating decimal number ,divide repeating digits with 10 raised to the power number of digits repeating minus one to get your desired answer.
Formula:
 $ fraction \times ({10^{no\,of\,digits\,repeating}} - 1) = {10^{no\,of\,digits\,repeating}} \times decimal\,no - \,decimal\,no $

Complete step-by-step answer:
In this question we are given a repeating decimal number i.e. $ 0.848484........ $ and we have to convert into a fraction.
So, in order to do so the first step is to eliminate the infinitely repeating digits by calculating the no of digits repeating after the decimal number and multiplying with the decimal number.
Here in our given decimal number only 2 digits are repeating forever after decimal point i.e. 84
So now we are multiplying the decimal with the 10 raised to the power no of digits repeating after decimal point i.e. 2 to eliminate the repeating part.
 $
   = 0.848484..... \times {10^2} \\
   = 0.848484..... \times 100 \\
   = 84.8484...... \;
  $
Now subtracting with the original number
 $ 84.8484........ - 0.8484....... = 84.0000 $
We know that our original decimal number is equal to some fraction, but we don’t yet know what that fraction is .
However, we do know that when it is subtracted from 100 times itself, we get $ 84 $
Let’s write this statement in an equation.
 $ 100 \times (fraction) - fraction = 84 $
It is possible to write this statement more simply, subtracting one thing from 100 thing leaves 99 things.
 $
  fraction \times (100 - 1) = 84 \\
  fraction \times 99 = 84 \;
  $
Now multiplying both sides by $ 99 $
 $ \dfrac{{fraction \times 99}}{{99}} = \dfrac{{84}}{{99}} $
Eliminating $ 99 $ from LHS, now our fraction becomes
 $ fraction = \dfrac{{84}}{{99}} $
Simplifying more, we get
 $ fraction = \dfrac{{28}}{{33}} $
Therefore, we have successfully converted our repeating decimal number $ 0.848484........ $ into $ fraction = \dfrac{{28}}{{33}} $ .
So, the correct answer is “ $ \dfrac{{28}}{{33}} $ .”.

Note: In algebra, a decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point. The dot in a decimal number is called a decimal point. The digits following the decimal point show a value smaller than one.