Consider the operation "minus the reciprocal of," defined by a\diamond b=a-(1)/(b). What is ((1\diamond2)\diamond3)-(1\diamond(2\diamond3))?

2015 AMC 10B Problem 7 — Solution

2015 AMC 10B Problem 7

Below is the professionally curated solution for Problem 7 of the 2015 AMC 10B, 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.

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Concepts:custom operation fraction

Difficulty rating: 960

Consider the operation "minus the reciprocal of," defined by $a\diamond b=a-\frac{1}{b}.$ What is $((1\diamond2)\diamond3)-(1\diamond(2\diamond3))?$

$-\dfrac{7}{30}$

$-\dfrac{1}{6}$

$0$

$\dfrac{1}{6}$

$\dfrac{7}{30}$

Solution:

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

Thus, the correct answer is A .

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