Question

If \[ 10.\overline{36}=\frac{p}{q}, \qquad \gcd(p,q)=1 \] then \(p+q=\)

• A. \(120\)
• B. \(124\)
• C. \(125\)
• D. \(130\)

Hint:

For repeating decimals, write the repeating block over as many 9’s as the number of repeating digits.

Answer 1

The Correct Option is C

Concept: For recurring decimals: \[ 0.\overline{ab}=\frac{ab}{99} \]

Step 1: Separate integer and decimal parts.
Given: \[ 10.\overline{36}=10+0.\overline{36} \] Now: \[ 0.\overline{36}=\frac{36}{99} \] Simplify: \[ =\frac{4}{11} \] So: \[ 10.\overline{36}=10+\frac{4}{11} \]

Step 2: Convert into improper fraction.
\[ =\frac{110+4}{11} \] \[ =\frac{114}{11} \] Thus: \[ p=114,\quad q=11 \] and: \[ \gcd(114,11)=1 \]

Step 3: Find \(p+q\).
\[ 114+11=125 \] Thus, the required answer is: \[ \boxed{125} \]