2022 AMC 10A Problem 1

P1-10~1 minPrint

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

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

Show solution without answering

Key insight

Work from the inside out. Each layer is 3 plus a fraction: add, then flip the result upside down to get the fraction for the next layer up.

Solution

The fraction is built in layers. Only the innermost layer can be worked out right away, so start there and move outward.

Innermost layer: $3 + \frac13 = \frac{9}{3} + \frac13 = \frac{10}{3}$ .
The layer above divides $1$ by this. Dividing $1$ by a fraction just turns it upside down: $\frac{1}{10/3} = \frac{3}{10}$ .
Middle layer: $3 + \frac{3}{10} = \frac{30}{10} + \frac{3}{10} = \frac{33}{10}$ .
Turn it upside down again: $\frac{1}{33/10} = \frac{10}{33}$ .
Outer layer: $3 + \frac{10}{33} = \frac{99}{33} + \frac{10}{33} = \frac{109}{33}$ .

Quick check: the answer should be a little more than $3$ , and $\frac{109}{33} \approx 3.3$ .

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

Why this works

Each layer of a stacked fraction like this is "add $3$ , then flip." You cannot finish an outer layer until you know the one under it, so always begin at the bottom. Keep every result as a fraction rather than a decimal, so each flip stays exact.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Number Theory, Chapter 11: Decimals and Fractions
Prealgebra, Chapter 6: Decimals (§6.3, §6.4)