Question

How do you convert 9.7 (7 being repeating) to a fraction?

Answer 1

Hint: Write the given repeating decimal as \[9.\overline{7}=9.777......\]. Assume this equation as x = 9.777…… and consider it is equation (1). Now, multiply both the sides with 10 and consider it as equation (2). Subtract equation (1) from equation (2) and divide both sides of the obtained difference with 9 to get the value of x in fractional form.

Complete step by step answer:
Here, we have been provided with the decimal number 9.7 in which 7 is repeating, this means we have been provided with the decimal number \[9.\overline{7}\]. We are asked to write it in the fractional form.
Now, since 7 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,
\[\Rightarrow x=9.\overline{7}\]
Removing the bar sign, we have,
\[\Rightarrow x=9.777......\] - (1)
Multiplying both the sides with 10, we get,
\[\Rightarrow 10x=97.777.......\] - (1)
Subtracting equation (1) from equation (2), we get,
\[\begin{align}
  & \Rightarrow 9x=88.000..... \\
 & \Rightarrow 9x=88 \\
\end{align}\]
Dividing both sides with 9, we get,
\[\Rightarrow x=\dfrac{88}{9}\]
Hence, \[\dfrac{88}{9}\] represents the fractional form of the decimal number \[9.\overline{7}\].

Note:
One may 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. That main thing that you have to remember is that somehow we have to make the digits after the decimal equal to 0.