2015 AMC 10B Problem 7

Problem 7 of 25EasierAlgebra

Consider the operation “minus the reciprocal of,” defined by

a\diamond b=a-\frac{1}{b}.

What is

((1\diamond2)\diamond3)-(1\diamond(2\diamond3))?

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Solution

\begin{aligned} &((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) \\ &= \left(\dfrac 12- \dfrac 13\right) - \left(1 \diamond \dfrac 53\right)\\ &= \left(\dfrac 12 - \dfrac 13\right) - \left(1-\dfrac 35\right)\\ &= \dfrac 16 - \dfrac 25 \\&= -\dfrac{7}{30} \end{aligned}

Thus, the correct answer is A .