2010 AMC 10A Problem 6

Below is the professionally curated solution for Problem 6 of the 2010 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 10A solutions, or check the answer key.

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Concepts:custom operation fraction

Difficulty rating: 1020

For positive numbers $x$ and $y$ the operation $\spadesuit (x,y)$ is defined as $\spadesuit (x,y) = x-\dfrac{1}{y}$ What is $\spadesuit (2,\spadesuit (2,2))?$

$\dfrac{2}{3}$

$1$

$\dfrac{4}{3}$

$\dfrac{5}{3}$

$2$

Solution:

Evaluating the inner expression, we get $\spadesuit (2, 2) = 2 - \dfrac{1}{2} = \dfrac{3}{2}.$ Then we have $\spadesuit \left(2, \dfrac{3}{2}\right) = 2 - \dfrac{1}{\frac{3}{2}} = \dfrac{4}{3}.$

Thus, C is the correct answer.

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