Skip to main content

2022 AMC 12A Problem 1

Problem 1 of 25EasierAlgebra

What is the value of 3+13+13+13?3+\cfrac{1}{3+\cfrac{1}{3+\frac13}}?

Answer choices

Show solution

Solution

Simplify from the bottom. The innermost fraction is 3+13=103.3+\dfrac13=\dfrac{10}{3}. The next layer is 3+1103=3+310=3310.3+\dfrac{1}{\frac{10}{3}}=3+\dfrac{3}{10}=\dfrac{33}{10}. Finally, 3+13310=3+1033=10933.3+\dfrac{1}{\frac{33}{10}}=3+\dfrac{10}{33}=\dfrac{109}{33}. Thus, the correct answer is D.

More practice

Concepts: continued fraction · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.