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2010 AMC 10B Problem 9

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2010 AMC 10B Problem 9AMC 10 Step by Step
Name:                             Date: September 13, 2026
Algebraic Manipulation · difficulty 2 of 5 · about 2 min · position P1-10

Lucky Larry's teacher asked him to substitute numbers for aa , bb , cc , dd , and ee in the expression a−(b−(c−(d+e)))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 aa , bb , cc , and dd were 11 , 22 , 33 , and 44 , respectively. What number did Larry substitute for ee ?

  1. A)

    −5-5

  2. B)

    −3-3

  3. C)

    00

  4. D)

    33

  5. E)

    55

Answer

D

Key insight

Simplifying gives a-b+c-d-e, while Larry computed a-b-c-d+e; the two agree exactly when e = c.

Solution

Remove the parentheses from the inside out, flipping signs each time a minus sign is distributed:
c−(d+e)=c−d−e,b−(c−d−e)=b−c+d+e,c - (d+e) = c - d - e,\qquad b - (c - d - e) = b - c + d + e,
a−(b−c+d+e)=a−b+c−d−e.a - (b - c + d + e) = a - b + c - d - e.

Larry, ignoring parentheses, computed a−b−c−d+ea - b - c - d + e.

The two results are equal when
a−b+c−d−e=a−b−c−d+e⟹2c=2e⟹e=c.a - b + c - d - e = a - b - c - d + e \quad\Longrightarrow\quad 2c = 2e \quad\Longrightarrow\quad e = c.
Since c=3c = 3, Larry used e=3e = 3. (Check: the correct value is 1−2+3−4−3=−51 - 2 + 3 - 4 - 3 = -5, and Larry's is 1−2−3−4+3=−51 - 2 - 3 - 4 + 3 = -5.)

The answer is (D) 3\boxed{\textbf{(D)}\ 3}.

Why this works

Comparing two expressions symbolically, before substituting numbers, reveals that most terms are identical and only cc and ee change sign. Simplifying first turns a five-variable problem into the one-line equation e=ce = c.

Alternative approach

Substitute the known values immediately: correct value 1−(2−(3−(4+e)))=−2−e1 - (2 - (3 - (4+e))) = -2 - e, Larry's value 1−2−3−4+e=e−81 - 2 - 3 - 4 + e = e - 8. Setting −2−e=e−8-2 - e = e - 8 gives e=3e = 3.

The trap

Losing track of the sign flips when removing nested parentheses, especially the double negative that makes +c.

Common mistakes

  • Losing track of the sign flips when removing nested parentheses, especially the double negative that makes +c.
  • Reading Larry's version as a−b−c−d−ea - b - c - d - e (with ee subtracted), which forces c=0c = 0 and matches no choice.

Techniques

Set up the equation/formula and compute; no special trick needed