Question

How do you convert $.63$ repeating to a fraction ?

Answer 1

$0.\overline{63}=\frac{7}{11}$ and $0.6\overline{3}=\frac{19}{30}$
Explanation :
I will use bar notation to indicate repeating digits, placing a bar over the sequence of digits which is repeated...
$0.63636363$.... $=0.\overline{63}$
If we multiply this repeating decimal by $(100-1)$ then we will get an integer: Multiplying by $100$ shifts the number left by $2$ places (the length of the repeating pattern), then we subtract the original number to cancel out the repeating tail...
$(100-1)0.\overline{63}=63.\overline{63}=0.\overline{63}=63$
Then divide both sides by $(100-1)$ and simplify :
$0.\overline{63}=\frac{63}{100-1}=\frac{63}{99}=\frac{7\cdot 9}{11\cdot 9}=\frac{7}{11}$
On second thoughts, may be you meant $0.6\overline{3}$, in which case.
Multiply by $10(10-1)$ to get an integer. The first $10$ shifts the number left one place, to leave the repeating pattern starting just after the decimal point. The $(10-1)$ shifts the number one more place (the length of the repeating pattern) to the left, then subtracts the original to cancel out the repeating tail...
$10(10-1)0.6\overline{3}=63.\overline{3}-6.\overline{3}=57$
Then divide both sides by $10(10-1)=100-10=90$ and simplify :
$0.6\overline{3}=\frac{57}{10(10-1)}=\frac{57}{90}=\frac{19\cdot 3}{30\cdot 3}=\frac{19}{30}$