The number 0.121212 ...... in the form $\frac{p}{q}$ is equal to

## Concept & Formula Converting a pure recurring decimal to a rational fraction requires algebraic manipulation where you multiply by a power of 10 to shift the decimal and subtract out the repeating part. $$ 0.\overline{xy} = \frac{xy}{99} $$ ## Step-by-Step Solution * **Calculation:** Let $x = 0.121212...$ (Equation 1) Since there are two repeating digits (12), multiply both sides by 100 to shift the decimal point two places to the right: $100x = 12.121212...$ (Equation 2) Subtract Equation 1 from Equation 2: $100x - x = 12.121212... - 0.121212...$ $99x = 12$ Solve for $x$: $x = \frac{12}{99}$ Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (3): $x = \frac{4}{33}$ ## Exam Strategy & Shortcut Memorize the recurring decimal shortcut rule: Write the repeating sequence as the numerator. Write as many '9's in the denominator as there are digits in the repeating sequence. Here, the repeating part is "12" (2 digits). Fraction = $\frac{12}{99}$. Divide top and bottom by 3 in your head $\implies \frac{4}{33}$. ## Common Pitfall Students often place the repeating digits over factors of 100 instead of 99, yielding $\frac{12}{100} = \frac{3}{25}$. A power of 10 in the denominator is only for terminating decimals, not recurring ones. ## Final Answer **Therefore, the correct answer is 4/33.**