$0.\overline{36}$ expressed in the form $\frac{p}{q}$ equals

Concept & Formula To express a pure recurring decimal in the form $\frac{p}{q}$, assign the repeating pattern as the numerator and place an equivalent number of $9$s as the denominator, then simplify the resulting fraction to its lowest terms. $$ 0.\overline{xy} = \frac{xy}{99} $$ Step-by-Step Solution Given the decimal expression: $0.\overline{36}$ The repeating part is $36$, which has two digits. We place $36$ over $99$. $$ \frac{p}{q} = \frac{36}{99} $$ Next, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $9$. Numerator: $\frac{36}{9} = 4$ Denominator: $\frac{99}{9} = 11$ $$ \frac{p}{q} = \frac{4}{11} $$ Exam Strategy & Shortcut Recognize that $36$ and $99$ are both multiples of $9$. You can quickly mentally reduce $\frac{36}{99}$ by dividing by $9$ to arrive straight at $\frac{4}{11}$. Common Pitfall Students often stop at $\frac{36}{99}$ and panic when they don't see it in the options, or worse, they incorrectly guess a similar-looking option like $\frac{35}{99}$. Always remember to reduce your fraction to its simplest form. Final Answer Therefore, the correct answer is $\frac{4}{11}$.