Question

How do you convert 0.87 (7 repeating) to a fraction?

Answer 1

Hint: Write the given repeating decimal as \[0.8\overline{7}=0.87777.....\]. Assume this equation as x = 0.87777….. and multiply both the sides with 10, consider it as equation (1). Now, multiply both the sides of x = 0.87777…… with 100 and consider it as equation (2). Subtract equation (1) from equation (2) and divide both sides of the obtained difference with 90 to get the value of x in fractional form.

Complete step by step answer:
Here, we have been provided with the decimal number 0.87 in which 7 is repeating, that means we have been provided with the decimal number \[0.8\overline{7}\]. We are asked to write it in the fractional form.
Now, since 7 will repeat up to infinite places after the decimal 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,
\[\Rightarrow x=0.8\overline{7}\]
Removing the bar sign, we have,
\[\Rightarrow x=0.87777.....\]
Multiplying both sides with 10, we get,
\[\Rightarrow 10x=8.7777.....\] - (1)
Now, multiplying both sides of the expression x = 0.87777….. with 100, we get,
\[\Rightarrow 100x=87.7777.....\] - (2)
Subtracting equation (1) from equation (2), we get,
\[\begin{align}
  & \Rightarrow 90x=\left( 87.777..... \right)-\left( 8.777..... \right) \\
 & \Rightarrow 90x=79.000..... \\
 & \Rightarrow 90x=79 \\
\end{align}\]
Dividing both sides with 90, we get,
\[\Rightarrow x=\dfrac{79}{90}\]
Hence, \[\dfrac{79}{90}\] represents the fractional form of the decimal number \[0.8\overline{7}\].

Note:
One may note that this given number is a rational number (non – terminating repeating) and that is why we were able to convert it in the fractional form. Note that we can also get the answer by subtracting x from 10x and then simplifying. In this case we would not be required to form equation (2) but the equation that will be obtained would be 9x = 7.9. Further we will multiply both sides with 10 and divide with 90. So, overall we will be doing the same thing. The main thing that you have to remember is that somehow we have to make the digits after decimal equal to 0.