Question

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

Answer 1

Step 1: Simplify by converting $0.87$ $(7$repeating$)$ to a fraction.

The decimal number $0.87$, in which the number $7$ repeats, is equivalent to giving us the decimal number $0.8\overline{7}$.

Consider the given decimal number as $x$.

$x=0.8\overline{7}$
Now, remove the bar sign:
$x=0.87777\dots ..$
Multiply with $10$ on both sides:
$10x=8.7777\dots \dots ...\left(1\right)$
The value is obtained by multiplying both sides of the expression $x=0.87777\dots ..$ by $100$:
$100x=87.7777\dots \dots ...\left(2\right)$

Step 2: Find the value of $x$.
Subtract equation $\left(1\right)$from equation $\left(2\right)$:

$100x-10x=87.7777\dots \dots -8.7777......$
$\begin{array}{rcl}& \Rightarrow & 90x=79.000\dots .\\ & \Rightarrow & 90x=79\end{array}$

By dividing both sides by $90$:

$\therefore x=\frac{79}{90}$

Therefore, $\frac{79}{90}$is the fractional representation of the decimal number $0.8\overline{7}$.