The Discount Double-Check
A jacket is priced at $80. A store advertises "25% off, then an additional 10% off the reduced price." A shopper calculates 25% + 10% = 35% off, so she expects to pay $80 − $28 = $52. At the register she is charged a different amount. She also has a coupon for $5 off any purchase over $50, which the cashier says must be applied after the percentage discounts. The shopper wants to know whether the store made an error, and she wants a general rule for combining percent discounts.
1. Compute the correct final price after both percentage discounts and determine whether the shopper's $52 expectation was right.
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Apply discounts sequentially, not additively. First discount: 25% of $80 = $20, so price = $60. Second discount: 10% of $60 = $6, so price = $54. The shopper expected $52, so she was wrong by $2 - the store did not err. Why adding fails: the second 10% is taken off the already-reduced $60, not the original $80. The combined multiplier is 0.75 × 0.90 = 0.675, so she pays 67.5% of $80 = $54. Adding 25% + 10% = 35% would correspond to a multiplier of 0.65, which overstates the discount. General rule: successive percent discounts multiply, and the combined percent off is 1 − (product of remaining fractions), here 1 − 0.675 = 32.5%, not 35%.
2. Apply the $5 coupon and explain how order of operations changes the outcome.
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Coupon after percentages: $54 − $5 = $49. Check the threshold: the purchase is over $50 before the coupon ($54), so it qualifies. If the coupon were applied first: $80 − $5 = $75, then 25% off = $56.25, then 10% off = $50.63. That is more than $49, so order matters. The store's stated order (percentages first) favors the shopper here because the $5 is subtracted from a smaller base but is a fixed dollar amount, so it removes a larger share. Key idea: percent discounts and fixed-dollar discounts do not commute. A fixed-dollar coupon is worth more, proportionally, when applied last to a small total; a percent discount applied first shrinks the base the coupon acts on.