Evaluate the continued fraction: x = 2 + 13 + 14 + 12

Question

Evaluate the continued fraction: x = 2 + 13 + 14 + 12

1. A. 67/29 (Correct answer)
2. B. 41/17
3. C. 45/19
4. D. 47/20

Correct Answer

Option A — 67/29

Detailed Solution & Explanation

The correct answer is 67/29.

Key Points

• Continued fractions are evaluated from the innermost denominator outwards.
• Innermost: $4 + \tfrac{1}{2} = \tfrac{9}{2}$.
• Next level: $3 + \dfrac{1}{9/2} = 3 + \tfrac{2}{9} = \tfrac{29}{9}$.
• Finally: $x = 2 + \dfrac{1}{29/9} = 2 + \tfrac{9}{29} = \tfrac{67}{29}$.

Additional Information

• Dividing by a fraction means multiplying by its reciprocal — that single step is where most errors occur.
• Working outward in one direction avoids ever holding a nested expression in your head.
• A quick sanity check: $67/29 \approx 2.31$, which is a little above 2, exactly as the leading term suggests.

प्रश्न (हिन्दी में)

सतत भिन्न का मूल्यांकन करें: x = 2 + 13 + 14 + 12

1. A. 67/29
2. B. 41/17
3. C. 45/19
4. D. 47/20