2009 AMC 10A Problem 3

Attempt Problem 3 of the 2009 AMC 10A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2009 AMC 10A solutions, or check the answer key.

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3.

Which of the following is equal to

1 + \cfrac{1}{1 + \cfrac{1}{1 + 1}}?

\dfrac{5}{4}

\dfrac{3}{2}

\dfrac{5}{3}

2

3

Answer: C

Concepts:continued fraction fraction

Difficulty rating: 870

Small hint:

Simplify from the innermost fraction outward

Big hint:

The inner expression

1 + \dfrac{1}{1+1}

equals

\dfrac{3}{2}

Solution:

Working outward,

\begin{aligned} &1 + \cfrac{1}{1 + \cfrac{1}{1 + 1}} \\ &= 1 + \cfrac{1}{1 + \dfrac{1}{2}} \\ &= 1 + \dfrac{1}{\frac{3}{2}} \\ &= 1 + \dfrac{2}{3} = \dfrac{5}{3}. \end{aligned}

Thus, the correct answer is C.