AMC 10 · 2010 · #9

Grade 7 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
order-of-operationslinear-equations-one-var convert-to-algebra ↑ Prerequisites: order-of-operations
📏 Medium solution 💡 2 insights
Problem
Larry must substitute numbers for a, b, c, d, and e in the expression a-(b-(c-(d+e))) and evaluate it. He drops every set of parentheses but keeps each + and - sign exactly where it is written, then adds and subtracts left to right. By luck, his stripped-down answer matches the true value of the full expression. The numbers he substituted for a, b, c, and d were 1, 2, 3, and 4, respectively. What number did Larry substitute for e?

Pick an answer.

(A)
-5
(B)
-3
(C)
0
(D)
3
(E)
5

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Keep e as an unknown letter and turn both computations into expressions in e (Tool #4, Introduce a Variable). Tool #7 (Identify Subproblems) handles the correct expression one layer at a time, peeling the nested parentheses from the inside out so the minus signs are distributed without error. Then Tool #13 (Convert to Algebra) turns the sentence "Larry's answer equals the correct answer" into a single equation in e, which solves in one line. Carrying e symbolically is cleaner than testing all five choices, because it produces the value directly.

1STEP 1

Expand the correct expression

Peel a-(b-(c-(d+e))) from the inside out, flipping signs behind each minus: c-d-e, then b-c+d+e, then a-b+c-d-e.

a-(b-(c-(d+e))) = a-b+c-d-e
2STEP 2

Write Larry's stripped version

Larry keeps every sign where it stands and just drops the brackets, reading straight across: a-b-c-d+e.

a-b-c-d+e
3STEP 3

Substitute the known numbers

Substitute a=1, b=2, c=3, d=4 and keep e a letter: the true value is -2-e and Larry's is -8+e.

correct=-2-e, Larry=-8+e
4STEP 4

Set them equal and solve

The coincidence means -2-e=-8+e, so 6=2e and e=3; both sides then equal -5.

-2-e=-8+e → 6=2e → e=3
Answer
3
Plug e=3 back into the original problem. Correct: 1-(2-(3-(4+3)))=1-(2-(3-7))=1-(2-(-4))=1-6=-5. Larry: 1-2-3-4+3=-5. Both give -5, so the coincidence really happens and e=3, choice (D), is right.
💡Key takeaway

A minus sign in front of a group flips every sign inside it, so expand the real expression carefully, write Larry's no-parentheses version, set the two equal, and solve for e to get 3.

  • Expand the correct expression
  • Write Larry's stripped version
  • Substitute the known numbers
  • Set them equal and solve

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