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
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.

Pick an answer.

(A)
6
(B)
9
(C)
10
(D)
12
(E)
15

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve an Easier Related Problem

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.

1STEP 1

A 50%-off sale means the sale price is half the regular price, so double 3togetthehalf−pound′sregularprice,<spanclass="mk−a"><spanclass="hl−ask">3 to get the half-pound's regular price, <span class="mk-a"><span class="hl-ask">6.

regular price (half pound) = 2 × 3=3 =6
2STEP 2

A full pound is two half-pounds, so double 6oncemoretoreachtheregularfull−poundprice,<spanclass="mk−a"><spanclass="hl−ask">6 once more to reach the regular full-pound price, <span class="mk-a"><span class="hl-ask">12 → (D).

regular price (full pound) = 2 × 6=6 =12 → (D)
Answer
12
Double-check with two halvings: starting from the answer 12perpound,halfapoundregularlycosts12 per pound, half a pound regularly costs6, and 50% off that is 3—exactlythesalepriceonthesign.Thearithmeticround−trips,so3 — exactly the sale price on the sign. The arithmetic round-trips, so12 is consistent. The other choices fail this check (e.g., 6perpoundwouldmakethesalehalf−poundonly6 per pound would make the sale half-pound only1.50, and 15perpoundwouldmakeit15 per pound would make it3.75).
💡Key takeaway

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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