2016 AMC 8 Problem 10

Attempt Problem 10 of the 2016 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 8 solutions, or check the answer key.

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

10.

Suppose that a∗ba \ast b means 3a−b.3a - b. What is the value of xx if 2∗(5∗x)=1?2 \ast (5 \ast x) = 1?
110 \dfrac{1}{10}
2 2
103 \dfrac{10}{3}
10 10
14 14
Answer: D
Concepts:custom operationlinear equation
Small hint:
Start by simplifying the inner operation 5∗x5\ast x.
Big hint:
Apply the definition a∗b=3a−ba\ast b=3a-b twice, from the inside out.
Video solution:
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Written solution:
First, 5∗x=15−x5\ast x=15-x. Applying the operation again gives 2∗(15−x)=6−(15−x)=x−9.\begin{aligned} 2\ast(15-x)&=6-(15-x)\\ &=x-9. \end{aligned} Thus x−9=1x-9=1, so x=10x=10.
Thus, D is the correct answer.
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