AMC 8 · 2013 · #2

Grade 6 arithmeticrate-ratio

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

percentageratio-proportionfraction-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic

📏 Short solution 💡 2 insights

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Problem

A sign at the fish market says, "50 $\%$ off, today only: half-pound packages for just $3 per package." What is the regular price for a full pound of fish, in dollars?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 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.

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Toolkit + CCSS Solution

Understand

Restated: The fish market is running a "$50\%$ off, today only" sale: a half-pound package costs $\$3$ on sale. We want the regular (non-sale) price of a full pound of fish, in dollars.

Givens: Sale discount $= 50\%$ off the regular price; Sale price of a half-pound package $= \$3$; Answer choices: (A) $6$, (B) $9$, (C) $10$, (D) $12$, (E) $15$ (dollars)

Unknowns: The regular (pre-discount) price of a $1$-pound amount of fish, in dollars

Understand

Restated: The fish market is running a "$50\%$ off, today only" sale: a half-pound package costs $\$3$ on sale. We want the regular (non-sale) price of a full pound of fish, in dollars.

Givens: Sale discount $= 50\%$ off the regular price; Sale price of a half-pound package $= \$3$; Answer choices: (A) $6$, (B) $9$, (C) $10$, (D) $12$, (E) $15$ (dollars)

Plan

Primary tool: #9 Solve an Easier Related Problem

Secondary: #8 Analyze the Units

The question asks about a full pound at the regular price, but the sign only tells us about a half pound at the sale price. Tool #9 (Easier Related Problem) says to first answer the smaller question we *can* answer — the regular price of a half pound — then scale up to a full pound. Tool #8 (Analyze the Units) keeps the bookkeeping straight: the $\$3$ is dollars per half-pound on sale, we need dollars per pound at regular price, so we have to undo the discount once and double the weight once.

Execute — Answer: D

#9 Solve an Easier Related Problem 6.RP.A.3 Step 1

Undo the $50\%$ discount to find the regular price of a half-pound package.
A $50\%$-off sale means the sale price is half of the regular price, so the regular price is twice the sale price.

$\text{regular price (half pound)} = 2 \times \$3 = \$6$

💡 Solving the easier half-pound version first is the Tool #9 move — once we know one half-pound costs $\$6$ regularly, the full pound is one short step away.

#8 Analyze the Units 4.OA.A.2 Step 2

Scale from half a pound up to a full pound.
A full pound is two half-pounds, so the regular price doubles again.

$\text{regular price (full pound)} = 2 \times \$6 = \$12 \;\Rightarrow\; \textbf{(D)}$

💡 Units check: $\dfrac{\$6}{\tfrac{1}{2}\text{ lb}} = \dfrac{\$12}{1\text{ lb}}$. Doubling the weight doubles the price at a constant per-pound rate.

[1] #9 6.RP.A.3 Undo the $50\%$ discount to find the regular price of a half-pound package. A $5

[2] #8 4.OA.A.2 Scale from half a pound up to a full pound. A full pound is two half-pounds, so

Review

Reasonableness: Double-check with two halvings: starting from the answer $\$12$ per pound, half a pound regularly costs $\$6$, and $50\%$ off that is $\$3$ — exactly the sale price on the sign. The arithmetic round-trips, so $\$12$ is consistent. The other choices fail this check (e.g., $\$6$ per pound would make the sale half-pound only $\$1.50$, and $\$15$ per pound would make it $\$3.75$).

Alternative: Tool #6 (Guess and Check) on the five choices: for each candidate regular full-pound price $P$, the sale half-pound price should be $\tfrac{P}{2} \times 0.5 = \tfrac{P}{4}$. Setting $\tfrac{P}{4} = 3$ gives $P = 12$, matching only choice (D). The other choices give $\tfrac{6}{4} = 1.50$, $\tfrac{9}{4} = 2.25$, $\tfrac{10}{4} = 2.50$, $\tfrac{15}{4} = 3.75$ — none equal $\$3$.

CCSS standards used (min grade 6)

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Scaling the half-pound price up to a full-pound price by doubling ($2 \times \$6 = \$12$), a Grade 4 "$n$ times as much" comparison.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including percent problems (Interpreting "$50\%$ off" as "sale price is half of regular," so regular price $= 2 \times$ sale price ($2 \times \$3 = \$6$ for the half pound).)

⭐ This AMC 8 problem just needs Grade 6 percent reasoning — "$50\%$ off" means double the sale price to get back to regular — plus one more doubling for the full pound.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 6)

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