Question
The number \(1.\overline{27}\) in the form \( \frac{p}{q} \), where \( q \neq 0 \), is:
(a) \( \frac{14}{9} \)
(b) \( \frac{14}{11} \)
(c) \( \frac{14}{13} \)
(d) \( \frac{14}{15} \)
Solution
Let \( x = 1.\overline{27} \)
\[ x = 1.272727\ldots \]
Multiply both sides by 100:
\[ 100x = 127.272727\ldots \]
Subtract the original equation:
\[ 100x – x = 127.2727\ldots – 1.2727\ldots \]
\[ 99x = 126 \]
\[ x = \frac{126}{99} = \frac{14}{11} \]
Final Answer
✔ Correct option: (b) \( \frac{14}{11} \)