Question
The number \(23.\overline{43}\) in the form \( \frac{p}{q} \), where \( q \neq 0 \), is:
(a) \( \frac{2320}{99} \)
(b) \( \frac{2343}{100} \)
(c) \( \frac{2343}{999} \)
(d) \( \frac{2320}{199} \)
Solution
Let \( x = 23.\overline{43} = 23.434343\ldots \)
Multiply by 100:
\[ 100x = 2343.4343\ldots \]
Subtract the original equation:
\[ 100x – x = 2343.4343\ldots – 23.4343\ldots \]
\[ 99x = 2320 \]
\[ x = \frac{2320}{99} \]
Final Answer
✔ Correct option: (a) \( \frac{2320}{99} \)