AMC 8 · 2014 · #1
Easy mode Grade 7Problem
Harry and Terry each get the same arithmetic problem: .
Harry does it the right way and follows the parentheses. He gets answer .
Terry forgets about the parentheses. He just writes and works left to right. He gets answer .
What is ?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: 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 $H-T$.
Givens: Harry's expression: $8-(2+5)$ (parentheses first); Terry's expression: $8-2+5$ (no parentheses, left to right); Answer choices: (A) $-10$, (B) $-6$, (C) $0$, (D) $6$, (E) $10$
Unknowns: The value of $H-T$
Understand
Restated: 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 $H-T$.
Givens: Harry's expression: $8-(2+5)$ (parentheses first); Terry's expression: $8-2+5$ (no parentheses, left to right); Answer choices: (A) $-10$, (B) $-6$, (C) $0$, (D) $6$, (E) $10$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #5 Look for a Pattern
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 $2 \times 5 = 10$. Spotting this pattern is a fast sanity check on the final $H-T$.
Execute — Answer: A
5.OA.A.1 Step 1 - Subproblem 1 — compute Harry's value $H$.
- The parentheses force $2+5$ to be evaluated first, then subtracted from $8$.
💡 Parentheses are a Grade 5 grouping symbol — do what is inside first, then continue.
5.OA.A.1 Step 2 - Subproblem 2 — compute Terry's value $T$.
- With no parentheses, only $+$ and $-$, work strictly left to right: $8-2$ first, then add $5$.
💡 Same Grade 5 standard, opposite trap: with no parentheses the $+5$ stays as addition, not subtraction.
7.NS.A.1 Step 3 - Subproblem 3 — combine the two results.
- Subtract $T$ from $H$; because $T>H$, the answer is negative.
💡 Subtracting a larger positive from a smaller one lands on the negative side of zero — Grade 7 integer subtraction.
5.OA.A.1 Subproblem 1 — compute Harry's value $H$. The parentheses force $2+5$ to be eval 5.OA.A.1 Subproblem 2 — compute Terry's value $T$. With no parentheses, only $+$ and $-$, 7.NS.A.1 Subproblem 3 — combine the two results. Subtract $T$ from $H$; because $T>H$, th Review
Reasonableness: 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 $2 \times 5 = 10$. That predicts $H-T = -10$, matching choice (A). Magnitude and sign both line up.
Alternative: Tool #5 (Look for a Pattern) used directly: dropping parentheses around $a+b$ in $x-(a+b)$ turns it into $x-a+b$, which differs by $+2b$. Here $b=5$, so the difference is $2 \times 5 = 10$ in Terry's favor and $H-T=-10$ without computing $H$ or $T$ individually.
CCSS standards used (min grade 7)
5.OA.A.1Use parentheses, brackets, or braces in numerical expressions and evaluate (Evaluating $H=8-(2+5)=1$ with the parenthesis rule, and evaluating $T=8-2+5=11$ left to right without parentheses.)7.NS.A.1Apply and extend understanding of addition and subtraction to rational numbers (Computing $H-T=1-11=-10$, which crosses zero into the negative integers.)
⭐ This AMC 8 problem only needs the Grade 5 parenthesis rule plus Grade 7 integer subtraction — no algebra at all.
⭐ This AMC 8 problem only needs the Grade 5 parenthesis rule plus Grade 7 integer subtraction — no algebra at all.
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