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

📘 View easy version →

Problem

Harry and Terry are each told to calculate $8-(2+5)$ . Harry gets the correct answer. Terry ignores the parentheses and calculates $8-2+5$ . If Harry's answer is $H$ and Terry's answer is $T$ , what is $H-T$ ?

Pick an answer.

(A)

-10

(B)

-6

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

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

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

#7 Identify Subproblems 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 $8$.

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

💡 Parentheses are a Grade 5 grouping symbol — do what is inside first, then continue.

#7 Identify Subproblems 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 $5$.

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

💡 Same Grade 5 standard, opposite trap: with no parentheses the $+5$ stays as addition, not subtraction.

#7 Identify Subproblems 7.NS.A.1 Step 3

Subproblem 3 — combine the two results.
Subtract $T$ from $H$; because $T>H$, the answer is negative.

$$H-T = 1-11 = -10 \;\Rightarrow\; \textbf{(A)}$$

💡 Subtracting a larger positive from a smaller one lands on the negative side of zero — Grade 7 integer subtraction.

[1] #7 5.OA.A.1 Subproblem 1 — compute Harry's value $H$. The parentheses force $2+5$ to be eval

[2] #7 5.OA.A.1 Subproblem 2 — compute Terry's value $T$. With no parentheses, only $+$ and $-$,

[3] #7 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.1 Use 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.1 Apply 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 7)

More like this