2018 AMC 10A Problem 1

Attempt Problem 1 of the 2018 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 2018 AMC 10A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

1.

What is the value of

\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?

\dfrac{5}{8}

\dfrac{11}{7}

\dfrac{8}{5}

\dfrac{18}{11}

\dfrac{15}{8}

Answer: B

Concepts:fraction order of operations

Difficulty rating: 560

Small hint:

Work from the innermost reciprocal outward

Big hint:

After each reciprocal step, simplify before adding the next

1

Solution:

We can simplify this as follows.

\begin{aligned} &\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\left(\dfrac{1}{3}+1\right)^{-1}+1\right)^{-1}+1 \\ &=\left(\dfrac{3}{4}+1\right)^{-1}+1 \\ &=\dfrac{4}{7} + 1 \\ &=\dfrac{11}{7} \end{aligned}

Thus, B is the correct answer.