Problem 16 · 1993 AJHSME
Hard
Fractions, Decimals & Percents
continued-fraction
11 + 12 + 13=
Show answer
Answer: C — 7/10.
Show hints
Hint 1 of 2
A stacked fraction like this can only be untangled from the innermost piece outward — you can't simplify the top until the bottom is a single number. Start at the deepest 2 + 1/3.
Still stuck? Show hint 2 →
Hint 2 of 2
Each layer is the same two-step move: add to make one fraction, then 'take 1 over it' (flip). Repeat that move as you climb out.
Show solution
Approach: peel the continued fraction from the inside out
- Deepest layer: 2 + 1/3 = 7/3. The next layer is 1 ÷ (7/3) = 3/7 — 'one over a fraction' just flips it.
- Climb out: 1 + 3/7 = 10/7. Then the outermost 1 ÷ (10/7) flips again to 7/10.
- Why this transfers: every continued (stacked) fraction unwinds bottom-up with the same rhythm — combine into one fraction, then flip when it's '1 over' it. Reciprocals (flipping) are the engine; reading it top-down would dead-end.
Mark:
· log in to save