2015 AMC 10B Problem 7

Attempt Problem 7 of the 2015 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2015 AMC 10B solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

7.

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))?

-\dfrac{7}{30}

-\dfrac{1}{6}

0

\dfrac{1}{6}

\dfrac{7}{30}

Answer: A

Concepts:custom operation fraction

Difficulty rating: 960

Small hint:

Compute

1\diamond2

and

2\diamond3

first

Big hint:

Remember

a\diamond b=a-\frac1b

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 .