Competition · AMC preparation · step 4 of 4
AMC 8 · 2013 · #2
Grade 6 arithmeticrate-ratioPick an answer.
AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Undo the half-price discount
A 50%-off sale means the sale price is half the regular price, so double $3 to get the half-pound's regular price, $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.
The regular price of a half-pound package is twice its sale price.
▸ Why?
The sale price is one half of the regular price, and doubling a half restores the whole, so the regular price is the sale price taken twice.
▸ Why?
A "fifty-percent off" sale means you pay fifty percent of the regular price, and fifty percent means fifty out of every hundred, which is exactly one of two equal halves of the whole.
▸ Why?
Charging only half the price divides the regular price by two, and multiplying by two reverses dividing by two, so doubling the sale price returns the full regular price.
Double up to a full pound
A full pound is two half-pounds, so double $6 once more to reach the regular full-pound price, $12 → (D).
Units check: 12/(1 lb). Doubling the weight doubles the price at a constant per-pound rate.
4.OA.A.2Analyze The UnitsThis 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.
- Undo the half-price discount
- Double up to a full pound
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