Question

How do you convert $0.63$ repeated to a fraction?

Answer 1

Hint:A repeating decimal is a decimal in which a digit or a block of digits repeats itself again and again. Such repeating decimals are represented by putting a bar on repeated digits or digits. In this given question we are required to convert a repeating decimal which is $0.6363....$ into a fraction. $0.6363....$ can be written as $0.\overline{6}\overline{3}$, putting a bar on repeating digits.

Complete step by step solution:
We need to assume that $x$=repeating decimal that is,
$x=0.6363....$ $0.\overline{6}\overline{3}$ ----(1)
Since, it is clearly visible that x is recurring in $2$ decimal places so we multiply $x$ by $100$.
$x\times 100=100\times 0.6363....$
$100x=63.6363....$ ----(2)
Now, we subtract equation (1) from equation (2)
$100x-x=63.6363....-0.6363....$
$99x=63$
The repeating digits $63$ cancel off and we get left with $99x$ being non-repeating.
Further, we divide left-hand side and right-hand side by $99$
${\displaystyle \frac{99x}{99}}={\displaystyle \frac{63}{99}}$
$x={\displaystyle \frac{63}{99}}$
Here, we got $x$ in the form of a fraction. In order to simplify the fraction, we divide $63$ by $99$.
$x={\displaystyle \frac{7}{11}}$
Therefore, the answer is $0.6363....={\displaystyle \frac{7}{11}}$

Note: Don’t confuse $0.63$ (repeating) means $0.6363....$Here in this question, we were given $0.63$ (repeating) which means both the digits are repeating and the bar is on both the digits. We should not assume that only one digit which is $3$ is repeating. If we assume so the question will totally change and become $0.6333....$ in this case, the bar would only be on digit $3$ which would look like $0.6\overline{3}$not on the digits $63$ which would look like $0.\overline{6}\overline{3}$.