Question

How do you write $0.\overline{86}$ as a fraction?

Answer 1

Hint: To write $0.\overline{86}$ as a fraction, we will first of all assume 0.868686…. as $x$ and then mark it as eq. (1) then multiply 100 on both the sides and mark this as eq. (2) then subtract eq. (1) from eq. (2) and then solve the subtraction to convert the recurring decimal to fraction.

Complete step-by-step answer:
The recurring decimal which is given in the above problem as follows:
0.86868686…….
Now, to convert above decimal into fraction we will assume the above recurring decimal as x then the above recurring decimal will look as follows:
$\Rightarrow x=0.868686....$ ……… Eq. (1)
Multiplying 100 on both the sides in the above equation we get,
$\Rightarrow 100x=86.8686........$ ………… Eq. (2)
Now, we are going to subtract eq. (1) from eq. (2) and we get,
$\begin{align}
  & 100x=86.8686..... \\
 & \dfrac{-x=-00.868686.......}{99x=86.000} \\
\end{align}$
Rewriting the above equation in the following way we get,
$\Rightarrow 99x=86$
Dividing 99 on both the sides we get,
$\Rightarrow \dfrac{99x}{99}=\dfrac{86}{99}$
On L.H.S of the above equation, 99 will be cancelled out from the numerator and the denominator and we get,
$\Rightarrow x=\dfrac{86}{99}$
In the above, as we have assumed x as the recurring decimal and in the above equation we have converted the decimal to a fraction and its value is $\dfrac{86}{99}$.

Note: There is a general pattern in which we can solve the recurring decimal problem which is shown above as follows:
The trick is we will multiply the decimal with the power of 10 in such a way so that the recurring number in the decimal will be cancelled out. Just like, in this problem, we have multiplied the given recurring decimal with 100 so that point will be shifted to two numbers 86 and then when we subtract this multiplication with the original recurring decimal then we have found that the recurring number “86” has been cancelled out after the decimal point.