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2009 AMC 10A Problem 3

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2009 AMC 10A Problem 3AMC 10 Step by Step
Name:                             Date: September 13, 2026
Fractions & Decimals · difficulty 1 of 5 · about 1 min · position P1-10

Which of the following is equal to 1+11+11+11 + \frac{1}{1+\frac{1}{1+1}} ?

  1. A)

    54\frac{5}{4}

  2. B)

    32\frac{3}{2}

  3. C)

    53\frac{5}{3}

  4. D)

    22

  5. E)

    33

Answer

C

Key insight

Simplify from the innermost fraction outward: 1 + 1/2 = 3/2, its reciprocal is 2/3, and 1 + 2/3 = 5/3.

Solution

Work from the inside out.

The innermost piece is 1+1=21 + 1 = 2, so the middle denominator is
1+12=32.1 + \frac{1}{2} = \frac{3}{2}.
Its reciprocal is 23\frac{2}{3}, so the whole expression is
1+23=53.1 + \frac{2}{3} = \frac{5}{3}.

The answer is (C) 53\boxed{\textbf{(C)}\ \frac{5}{3}}.

Why this works

A nested (continued) fraction is evaluated bottom-up: each layer is "one plus the reciprocal of the layer below." Doing one layer at a time, and writing each intermediate result as a single fraction, avoids losing track of which number is the numerator.

The trap

Flipping the wrong way, treating 1/(3/2) as 3/2 and getting 5/2, or stopping at 3/2 and choosing (B).

Common mistakes

  • Flipping the wrong way, treating 1/(3/2) as 3/2 and getting 5/2, or stopping at 3/2 and choosing (B).
  • Adding the 11's first as if the expression were 1+1+11 + 1 + 1 and answering 33.

Techniques

Set up the equation/formula and compute; no special trick needed