2022 AMC 10A Problem 1

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

1. Innermost layer: $3 + \frac13 = \frac{9}{3} + \frac13 = \frac{10}{3}$.
2. The layer above divides $1$ by this. Dividing $1$ by a fraction just turns it upside down: $\frac{1}{10/3} = \frac{3}{10}$.
3. Middle layer: $3 + \frac{3}{10} = \frac{30}{10} + \frac{3}{10} = \frac{33}{10}$.
4. Turn it upside down again: $\frac{1}{33/10} = \frac{10}{33}$.
5. 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 $\boxed{\textbf{(D)}\ \frac{109}{33}}$.

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.

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