Skip to main content

2010 AMC 10A Problem 6

Problem 6 of 25EasierAlgebraArithmetic

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

Answer choices

Show solution

Solution

Evaluating the inner expression, we get ♠(2,2)=2−12=32. \spadesuit (2, 2) = 2 - \dfrac{1}{2} = \dfrac{3}{2}. Then we have ♠(2,32)=2−132=43. \spadesuit \left(2, \dfrac{3}{2}\right) = 2 - \dfrac{1}{\frac{3}{2}} = \dfrac{4}{3}. Thus, C is the correct answer.
AoPS wiki

Tagged: custom operation · fraction

More practice