2010 AMC 10A Problem 6

Question

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

Evaluate the inner

\spadesuit

expression first

\spadesuit(2,2)=2-\dfrac12

, then use that as the second input

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.

Problem 6 in other years