AMC 8 · 2014 · #1

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

order-of-operationsmulti-digit-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticorder-of-operations

📏 Short solution 💡 2 insights

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Problem

Harry evaluates 8-(2+5) correctly, so he respects the parentheses. Terry drops the parentheses and instead evaluates 8-2+5 left to right. Call Harry's result H and Terry's result T — compute H-T.

Pick an answer.

(A)

-10

(B)

-6

(C)

(D)

(E)

AMC 8 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question packages three small calculations into one: compute H, compute T, then subtract. Tool #7 (Identify Subproblems) splits these so the parenthesis rule and the left-to-right rule do not get tangled. Tool #5 (Look for a Pattern) names the trap behind the problem — dropping parentheses around (2+5) flips the sign of the 5 inside, so T overshoots H by exactly 2 × 5 = 10. Spotting this pattern is a fast sanity check on the final H-T.

1STEP 1

Subproblem 1 — parentheses first: 2+5=7, then 8-7, so Harry's H = 1.

H = 8-(2+5) = 8-7 = 1

2STEP 2

Subproblem 2 — no parentheses, so go left to right: 8-2=6, then +5, giving Terry's T = 11.

T = 8-2+5 = 6+5 = 11

3STEP 3

Subproblem 3 — combine: T is larger, so H-T = 1-11 = -10, i.e. choice (A).

H-T = 1-11 = -10 → (A)

Answer

-10

Sign check: Terry's expression 8-2+5 replaces a subtraction-of-5 (inside Harry's parentheses) with an addition-of-5, so Terry's value must be larger than Harry's by 2 × 5 = 10. That predicts H-T = -10, matching choice (A). Magnitude and sign both line up.

💡Key takeaway

This AMC 8 problem only needs the Grade 5 parenthesis rule plus Grade 7 integer subtraction — no algebra at all.

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