AMC 10 · 2009 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

fraction-arithmeticorder-of-operations identify-subproblemswork-backwards ↑ Prerequisites: fraction-arithmetic

📏 Medium solution 💡 1 insight

Problem

Evaluate the nested (continued) fraction 1 + 1/(1 + 1/(1 + 1)), in which every fraction bar groups everything written under it, and match its value to one of the five answer choices.

Pick an answer.

(A)

$\frac {5}{4}$

(B)

$\frac {3}{2}$

(C)

$\frac {5}{3}$

(D)

(E)

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

A stacked fraction like this cannot be attacked from the top, because the top 1/… needs its denominator finished first. Tool #7 (Identify Subproblems) says: break the tower into layers and solve the smallest, innermost one first. Here the innermost subproblem is just 1 + 1. Once that is a number, the next fraction becomes computable, and so on outward. That inside-out order is also Tool #11 (Work Backwards) — the deepest layer is really the starting point. Tool #3 (Eliminate Possibilities) gives a quick sanity net: the outer "1 + (something between 0 and 1)" forces the answer to sit strictly between 1 and 2.

1STEP 1

Peel to the innermost layer

A fraction bar hides parentheses, so start at the bottom: the deepest denominator 1 + 1 is just 2.

1 + 1 = 2 → 1 + 1/(1 + 1/2)

2STEP 2

Climb up one floor

Next layer up: 1 + 1/2. Write 1 as 2/2 so the denominators match, and the sum is 3/2.

1 + 1/2 = 2/2 + 1/2 = 3/2

3STEP 3

Flip the middle fraction

The tower now reads 1 + 1 ÷ 3/2. Dividing by a fraction flips it, so 1 ÷ 3/2 = 2/3.

1/3/2 = 1 ÷ 3/2 = 1 · 2/3 = 2/3 → 1 + 2/3

4STEP 4

Add the outermost 1

Top layer: 1 + 2/3 = 3/3 + 2/3 = 5/3, which is choice (C).

1 + 2/3 = 3/3 + 2/3 = 5/3 → (C)

Answer

5/3

The outer part is 1 + (something between 0 and 1), since the inner fraction 2/3 is a proper fraction. So the answer must lie strictly between 1 and 2 — this kills (D) 2 and (E) 3 immediately. Among the survivors 5/4=1.25, 3/2=1.5, 5/3≈1.67, only 5/3 equals 1 + 2/3, confirming (C).

💡Key takeaway

A stacked fraction is a tower — start at the bottom, finish each denominator before you divide, and climb out one floor at a time to reach 5/3.

Peel to the innermost layer
Climb up one floor
Flip the middle fraction
Add the outermost 1

Peel to the innermost layer

Climb up one floor

Flip the middle fraction

Add the outermost 1

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