2010 AMC 12B Problem 5

Below is the professionally curated solution for Problem 5 of the 2010 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12B solutions, or check the answer key.

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Concepts:order of operations linear equation

Difficulty rating: 1100

Lucky Larry's teacher asked him to substitute numbers for $a, b, c, d,$ and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a, b, c,$ and $d$ were $1, 2, 3,$ and $4,$ respectively. What number did Larry substitute for $e?$

$-5$

$-3$

$0$

$3$

$5$

Solution:

The correct value is $a-(b-(c-(d+e)))=a-b+c-d-e.$ With $a, b, c, d=1, 2, 3, 4,$ this equals $1-2+3-4-e=-2-e.$

Larry dropped the parentheses and computed $1-2-3-4+e=-8+e.$

Setting $-2-e=-8+e$ gives $2e=6,$ so $e=3.$

Thus, the correct answer is D.

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