$1.\overline{27}$ in the form $\frac{p}{q}$ is equal to

Concept & Formula For a number with an integer part and a pure recurring decimal part, separate the integer from the decimal. Convert the recurring decimal to a fraction, simplify it, and then add it back to the integer. $$ a.\overline{xy} = a + \frac{xy}{99} $$ Step-by-Step Solution Given the expression: $1.\overline{27}$ We can split this into an integer and a decimal part: $1 + 0.\overline{27}$ Convert the pure recurring decimal $0.\overline{27}$ to a fraction: $$ 0.\overline{27} = \frac{27}{99} $$ Simplify $\frac{27}{99}$ by dividing numerator and denominator by $9$: $$ \frac{27}{99} = \frac{3}{11} $$ Now, add the integer part back to this fraction: $$ 1 + \frac{3}{11} = \frac{11 + 3}{11} = \frac{14}{11} $$ Exam Strategy & Shortcut Use the shortcut formula directly: $\frac{\text{Entire number} - \text{Non-repeating part}}{99} = \frac{127 - 1}{99} = \frac{126}{99}$. Dividing top and bottom by $9$ instantly yields $\frac{14}{11}$. Common Pitfall Many students mistakenly treat the bar as a standard decimal and write $\frac{127}{100}$ (Option a). Always pay close attention to the vinculum (the overline) which indicates a repeating sequence, necessitating a denominator of $99$. Final Answer Therefore, the correct answer is $\frac{14}{11}$.