AMC 8 · 2012 · #8

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagefraction-multiplication identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Short solution 💡 2 insights

Problem

A shop marks everything at half price. A coupon then takes an additional 20% off the sale price. What total percent off the original price does a shopper get by using the coupon during the sale?

Pick an answer.

(A)

$hspace{.05in}10$

(B)

$hspace{.05in}33$

(C)

$hspace{.05in}40$

(D)

$hspace{.05in}60$

(E)

$hspace{.05in}70$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce Variables

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.

1STEP 1

Let the original price be P, so every later price is just a fraction of P.

original price = P

2STEP 2

"Half price" makes the sale price 50% of P — multiply by 0.5 to get 0.5P.

sale price = P × (1 - 0.50) = 0.5P

3STEP 3

The 20% coupon comes off 0.5P, leaving 80% of it: 0.5P × 0.8 = 0.4P.

final price = 0.5P × (1 - 0.20) = 0.5P × 0.8 = 0.4P

4STEP 4

The final price 0.4P is 40% of P, so the shopper saved the other 60% off the original.

percent off = 100% - 40% = 60% → (D)

Answer

hspace{.05in}60

Try a concrete number: let P =

100. Half price makes it

50; a 20% coupon off

50 saves

10, leaving

40. The shopper paid

40 out of

100, a savings of

60 — exactly 60% off. A common trap is to add 50% + 20% = 70% (choice E), but the 20% only applies to the already-halved price, so the actual total is less than 70%.

💡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.

More like this