AMC 8 · 2013 · #2
Easy mode Grade 6Problem
Picture a sign at the fish market. It says: today only, half-pound packages are 50% off, and one half-pound package costs just $3.
That means $3 is the sale price for half a pound. The regular price (no discount) is twice as much per half-pound.
What is the regular price for a whole pound of fish, in dollars?
Pick an answer.
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.
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
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.
💡 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.
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.
💡 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.
6.RP.A.3 Undo the $50\%$ discount to find the regular price of a half-pound package. A $5 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.2Multiply 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.3Use 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.
⭐ 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.
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