AMC 10 · 2022 · #1

Grade 5 arithmetic

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

fraction-arithmeticfraction-multiplication identify-subproblemswork-backwards ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 1 insight

Problem

Evaluate the three-level continued fraction

3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}}

and match the value to one of the five answer choices.

Pick an answer.

(A)

$\frac{31}{10}$

(B)

$\frac{49}{15}$

(C)

$\frac{33}{10}$

(D)

$\frac{109}{33}$

(E)

$\frac{15}{4}$

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression has three nested layers, so we cannot evaluate it left-to-right. Tool #7 (Identify Subproblems) splits the giant fraction into three identical small jobs: "compute 3 + 1/(something already known)". Tool #11 (Work Backwards) tells us where to start — at the deepest layer, then climb outward, replacing each □ with the value we just found. Algebra (#13) is overkill since there are no unknowns; the layered structure tells us the path.

1STEP 1

Innermost layer: rewrite 3 as $\frac{9}{3}$ and add to $\frac{1}{3}$ , giving $\frac{10}{3}$ .

3 +

\frac{1}{3}

\frac{9}{3}

\frac{1}{3}

\frac{10}{3}

2STEP 2

Middle layer: dividing 1 by $\frac{10}{3}$ flips it to $\frac{3}{10}$ .

\frac{1}{\frac{10}{3}}

= 1 ÷

\frac{10}{3}

= 1 ×

\frac{3}{10}

\frac{3}{10}

3STEP 3

Finish the middle layer: 3 becomes $\frac{30}{10}$ , plus $\frac{3}{10}$ gives $\frac{33}{10}$ .

3 +

\frac{3}{10}

\frac{30}{10}

\frac{3}{10}

\frac{33}{10}

4STEP 4

Outer layer: flip $\frac{33}{10}$ to $\frac{10}{33}$ , add to 3 = $\frac{99}{33}$ , giving $\frac{109}{33}$ .

3 +

\frac{10}{33}

\frac{99}{33}

\frac{10}{33}

\frac{109}{33}

→ (D)

Answer

\frac{109}{33}

Sanity check the size. Each layer is roughly 3 + 1/(a bit more than 3), so the whole thing should be a little more than 3 but well under 4.

\frac{109}{33}

≈ 3.303 fits that exactly. Choice (C)

\frac{33}{10}

= 3.3 is the middle layer's value (a common trap if you stop one level too early); (D) is the genuine outer value, matching the answer.

💡Key takeaway

This AMC 10 problem only needs Grade 5 "add fractions with unlike denominators and flip a fraction" — work from the deepest $\frac{1}{3}$ outward, one layer at a time.

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