2011 AMC 10A Problem 3

Below is the professionally curated solution for Problem 3 of the 2011 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 10A solutions, or check the answer key.

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Concepts:custom operation mean fraction

Difficulty rating: 870

Suppose $[a b]$ denotes the average of $a$ and $b,$ and { $a$ $b$ $c$ } denotes the average of $a,$ $b,$ and $c.$ What is $\{\{1 \ 1 \ 0\} \ [0 \ 1] \ 0\}?$

$\dfrac{2}{9}$

$\dfrac{5}{18}$

$\dfrac{1}{3}$

$\dfrac{7}{18}$

$\dfrac{2}{3}$

Solution:

We have that $\{1 \ 1 \ 0\} = \dfrac{1 + 1 + 0}{3} = \dfrac{2}{3}.$

We also get that $[0 \ 1] = \dfrac{1 + 0}{2} = \dfrac{1}{2}.$

Finally, $\left\{\dfrac{2}{3} \ \dfrac{1}{2} \ 0\right\} = \dfrac{\dfrac{2}{3} + \dfrac{1}{2} + 0}{3} = \dfrac{7}{18}.$

Thus, D is the correct answer.

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