2010 AMC 10A Problem 6

Problem 6 of 25EasierAlgebra

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))?

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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.

Concepts: custom operation · fraction

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