Competition · AMC preparation · step 4 of 4

AMC 8 · 2002 · #14

Grade 7 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps
percentagefraction-multiplicationfraction-arithmetic identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic
📏 Short solution 💡 2 insights
Problem
A store first takes 30% off every item. Later it takes another 20% off those already-discounted prices. What is the total percent discount compared to the original price?

Pick an answer.

(A)
35%
(B)
44%
(C)
50%
(D)
56%
(E)
60%

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The original price is never given, so Tool #4 (Introduce a Variable) — let the original price be P — turns the abstract "percent of percent" question into clean arithmetic in P. Tool #7 (Identify Subproblems) splits the chain of discounts into two clear steps: first apply the 30% discount to get the sale price, then apply the 20% discount to that sale price. Comparing the final price to P at the end gives the total percent off.

1STEP 1

Name the original price

Let the original price be P, so we can track what fraction of P is left at each step.

original price = P
2STEP 2

Apply the first discount

A 30% discount leaves 70% of the price, so the sale price is 0.7P.

sale price = P × (1 - 0.30) = 0.7P
3STEP 3

Apply the second discount

Taking 20% off keeps 80% of the sale price, so the final price is 0.7P × 0.8 = 0.56P.

final price = 0.7P × (1 - 0.20) = 0.7P × 0.8 = 0.56P
4STEP 4

Compare to the original price

Since 0.56P is 56% of P, the customer paid 56% and saved 44% off the original.

total discount = 100% - 56% = 44% → (B)
Answer
44%
Try a concrete number: let P = $100. The 30% discount drops the price to $70; the 20% discount takes $14 off $70, leaving $56. The customer paid $56 out of $100, a savings of $44 — exactly 44% off. The common trap is to add 30% + 20% = 50% (choice C), but the 20% only applies to the already-reduced price, so the actual total is less than 50%. Also note choice (D) 56% is the percent paid, not the percent off.
💡Key takeaway

Stacked 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 first discount
  • Apply the second discount
  • Compare to the original price

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