Stacked Discounts at Two Competing Stores
A jacket has a list price of $180 at two competing stores. Store A advertises '20% off, then an extra 15% off the reduced price,' and the shopper also holds a $12 coupon that is applied after both discounts but before tax. Store B advertises a single discount of 32% off the same list price, with no coupon. Both stores charge 8% sales tax on the post-discount price. Store A is now considering replacing its 'extra 15% off' with a flat 'additional $30 off the reduced price' for a holiday promotion. The manager insists that stacking 20% and 15% is 'more generous than a single 32% off' and wants the arithmetic verified before the sale signage is printed.
1. Compute the final after-tax price at each store for the $180 jacket. Which store is cheaper, and by how much?
Read the worked answer
Store A (stacked discounts plus coupon)
- First discount: $180(0.80) = $144.00
- Second discount: $144.00(0.85) = $122.40
- Coupon: $122.40 − $12 = $110.40
- Tax: $110.40(1.08) = $119.23
Store B (single 32% discount)
- $180(0.68) = $122.40
- Tax: $122.40(1.08) = $132.19
Store A is cheaper by $132.19 − $119.23 = $12.96, which is exactly the $12 coupon plus its 8% tax ($12 × 1.08). Note that $0.80 × $0.85 = $0.68, so the two stores' discount structures are identical before the coupon - the entire gap comes from the coupon, not the stacking.
2. Is the manager correct that stacking 20% and 15% is more generous than a single 32% discount? Find the single discount rate equivalent to the stacked pair and identify the likely source of the manager's error.
Read the worked answer
Multiply the retained fractions: 0.80 × 0.85 = 0.68, so the customer pays 68% of list - a 32% total discount, exactly equal to Store B's single discount.
The manager is not correct: the two promotions are equivalent on identical list prices. The error is the common additive shortcut 20% + 15% = 35%, which overstates the discount by 3 percentage points. Percentages of a shrinking base cannot be added; only the multipliers can be multiplied.
Rule to carry forward: a chain of discounts d₁, d₂, … is equivalent to the single discount 1 − (1 − d₁)(1 − d₂)…
3. Store A compares two second-stage promotions on top of the 20% off: 'extra 15% off' versus 'additional $30 off.' For what list price do the two promotions produce the same price, and which is better for a customer below and above that price?
Read the worked answer
Let L be the list price. Both start from 0.80L.
- Extra 15%: 0.80L(0.85) = 0.68L
- Flat $30: 0.80L − 30
Set equal: 0.68L = 0.80L − 30 → 30 = 0.12L → L = $250.
- Below $250, the flat $30 removes a larger fraction than 15% of the reduced price, so it is better. Check L = $200: 0.68(200) = $136 vs. 160 − 30 = $130, so the flat amount wins.
- Above $250, the percentage wins. Check L = $300: 0.68(300) = $204 vs. 240 − 30 = $210, so 15% wins.
Judgement point: at the $180 list price in this problem, the flat $30 gives $114 vs. $122.40, a saving of $8.40 - a fact the manager's 'percentages are always bigger' intuition would miss.