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2022 AMC 10A Problem 1

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2022 AMC 10A Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Fractions & Decimals · difficulty 1 of 5 · about 1 min · position P1-10

What is the value of $3+\frac{1}{3+\frac{1}{3+\frac13}}?$

A)
$\frac{31}{10}$
B)
$\frac{49}{15}$
C)
$\frac{33}{10}$
D)
$\frac{109}{33}$
E)
$\frac{15}{4}$

Answer

Key insight

Evaluate the nested fraction from the inside out: 3+1/3 = 10/3, then 3+3/10 = 33/10, then 3+10/33 = 109/33.

Solution

Work from the innermost level outward, simplifying one layer at a time and keeping every result as a single fraction.

Innermost layer: $3 + \frac13 = \frac{10}{3}$ , whose reciprocal is $\frac{3}{10}$ .

Middle layer: $3 + \frac{3}{10} = \frac{33}{10}$ , whose reciprocal is $\frac{10}{33}$ .

Outer layer: $3 + \frac{10}{33} = \frac{99 + 10}{33} = \frac{109}{33}$ .

The answer is $\boxed{\textbf{(D)}\ \frac{109}{33}}$ .

Why this works

A continued fraction is just repeated "add, then take the reciprocal." The only workable order is inside-out, because each layer needs the value of the one beneath it. Keeping intermediate results as exact fractions (never decimals) makes each reciprocal a simple flip.

The trap

Working from the outside in, or flipping a fraction incorrectly (for example writing 1/(10/3) as 10/3 instead of 3/10).

Common mistakes

Working from the outside in, or flipping a fraction incorrectly (for example writing 1/(10/3) as 10/3 instead of 3/10).
Stopping one layer early and answering $\frac{33}{10}$ , choice (C), which is the value of the middle layer.

Techniques

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