2010 AMC 10B Problem 9

Problem 9 of 25EasierAlgebra

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?

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Solution

Ignoring the parentheses, Larry would get

1 - 2 - 3 - 4 + e = e - 8.

Evaluating with the parentheses, one would get

\begin{aligned} &1 - (2 -(3 - (4 + e)))\\ & = 1 - (2 - (-1 - e))\\ &= 1 - (3 + e) \\ &=-2 - e.\end{aligned}

Both of these values are the same, so

e - 8 = -2 - e

e = 3.

Thus, D is the correct answer.

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