Evaluate the continued fraction: x = 2 + 1⁄(3 + 1⁄(4 + 1⁄2))

Correct — A. Work from the bottom level up.

4 + 1⁄2 = 9⁄2
1 ÷ (9⁄2) = 2⁄9
3 + 2⁄9 = 29⁄9
1 ÷ (29⁄9) = 9⁄29

2 + 9⁄29 = 58⁄29 + 9⁄29 = 67⁄29 → option (a)

Why the others are wrong

• (b)
  41⁄17 — 41⁄17 = 2 + 7⁄17, and 7⁄17 ≈ 0.41 is above 1⁄3. But 1⁄(3 + a positive fraction) is always below 1⁄3, so x must stay under 2.33.
• (c)
  45⁄19 — 45⁄19 = 2 + 7⁄19 ≈ 2.37, above 2 + 1⁄3 ≈ 2.33. The part after 2 is 1⁄(3 + 2⁄9), which is below 1⁄3, so x cannot reach 2.37.
• (d)
  47⁄20 — 47⁄20 = 2.35, again above 2 + 1⁄3 ≈ 2.33. Since 3 + 2⁄9 is more than 3, its reciprocal is less than 1⁄3, and x is less than 2.33.

A continued fraction is evaluated from the innermost level outward: simplify the deepest denominator, take its reciprocal, add the whole number above it, and repeat.

For eliminating options, use a bound. The denominator 3 + 1⁄(4 + 1⁄2) lies between 3 and 4, so the part after 2 lies between 1⁄4 and 1⁄3, and x lies between 2.25 and 2.33.

Of the four options, 67⁄29 ≈ 2.31 is the one inside that range.

Key facts

• Evaluate a continued fraction from the innermost denominator outward.
• 1 ÷ (a⁄b) = b⁄a.
• 67⁄29 ≈ 2.310, while 2 + 1⁄3 ≈ 2.333.

Common traps

• Adding 1⁄2 to 4 and then forgetting to take the reciprocal before adding 3.
• Treating 1⁄(4 + 1⁄2) as 1⁄4 + 2, when a reciprocal does not split over a sum.

No directly related past PYQ was found.