Competition · AMC preparation · step 4 of 4
AMC 8 · 2012 · #8
Grade 7 arithmeticPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The original price is never stated, so Tool #5 (Introduce Variables) — let the original price be P — turns the abstract "percent of percent" question into concrete arithmetic. Tool #7 (Identify Subproblems) splits the chain of discounts into two clean steps: first apply the 50% sale to get the sale price, then apply the 20% coupon to that sale price. Comparing the final price to P at the end gives the total percent off.
Name the original price
Let the original price be P, so every later price is just a fraction of P.
Naming the unknown original price is the Tool #5 move and is exactly the Grade 6 idea of writing an expression with a letter standing for a number.
6.EE.A.2Look For A PatternApply the 50 percent sale
"Half price" makes the sale price 50% of P — multiply by 0.5 to get 0.5P.
Treating a 50% discount as multiplying by 0.5 is the Grade 6 percent-of-a-quantity move.
6.RP.A.3Identify SubproblemsApply the 20 percent coupon
The 20% coupon comes off 0.5P, leaving 80% of it: 0.5P × 0.8 = 0.4P.
Stacking a second percent discount on top of an already-discounted price is a Grade 7 multi-step percent-change problem.
Using the coupon on the already half-price item, the shopper ends up paying 0.4P — that is, the second discount lands on the reduced price 0.5P, not on the original P.
▸ Why?
Half price has already scaled the original P down to 0.5P, and the coupon takes 20% off that sale price; taking 20% off means keeping the other 80%, so what the shopper pays is 0.80 × 0.5P.
▸ Why?
20% means 20 out of every hundred, so removing 20% strips away 20/100 of the sale price and leaves 80/100 = 0.80 of it — that is exactly what turns "20% off" into multiplying by 0.8.
▸ Why?
Scaling P by 0.5 and then scaling that result by 0.8 gives the very same number as scaling P a single time by the product 0.5 × 0.8 = 0.4, which is why the two discounts combine into 0.4P instead of being added.
▸ Why?
In the product P × 0.5 × 0.8 you may multiply any two factors first, so grouping the two discount factors together gives P × (0.5 × 0.8) = P × 0.4.
Compare to the original price
The final price 0.4P is 40% of P, so the shopper saved the other 60% off the original.
Subtracting the "percent paid" from 100% to get the "percent off" is the standard Grade 7 percent-change wrap-up.
7.RP.A.3Identify SubproblemsStacked discounts multiply, not add — once you write the original price as P, this AMC 8 question is just Grade 7 percent reasoning.
- Name the original price
- Apply the 50 percent sale
- Apply the 20 percent coupon
- Compare to the original price
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