The Discount That Wasn't
A small bakery advertises: "Buy 2 dozen muffins, get the 3rd dozen 50% off - plus take 10% off your whole order with our loyalty card." A customer orders 3 dozen muffins at $9.00 per dozen and presents a loyalty card. The cashier applies the 50% discount to the third dozen first ($4.50 instead of $9.00), then takes 10% off the total. A second cashier insists the order should instead be: 10% off all three dozen first, then 50% off the already-discounted third dozen. The customer suspects the two methods give different totals and asks which is correct and whether the store's advertised "save up to 35%" claim holds.
1. Compute the total under each cashier's method and compare.
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Method 1 (50% off 3rd dozen, then 10% off total):
- Dozens 1–2: $9.00 + $9.00 = $18.00
- Dozen 3: $9.00 × 0.5 = $4.50
- Subtotal: $22.50
- Loyalty: $22.50 × 0.90 = $20.25
Method 2 (10% off all, then 50% off 3rd):
- Each dozen at 10% off: $9.00 × 0.90 = $8.10
- Dozens 1–2: $16.20
- Dozen 3: $8.10 × 0.5 = $4.05
- Total: $20.25
They are equal. Percentage discounts multiply, and multiplication is commutative: 0.5 × 0.9 = 0.9 × 0.5. The customer's suspicion is unfounded - order of applying percentage discounts doesn't change the total.
2. Test the "save up to 35%" claim against the actual savings.
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Full price: 3 × $9.00 = $27.00. Paid: $20.25. Savings: $27.00 − $20.25 = $6.75. Percent saved: $6.75 ÷ $27.00 = 0.25 = 25%.
The customer saved 25%, not 35%. The "up to 35%" claim would require a larger discount than advertised here. A 35% saving on $27.00 would be $9.45 off, leaving $17.55 - the order is $2.70 above that. The claim is not supported for this purchase.
3. A manager says, "Just add the discounts: 50% + 10% = 60% off." Explain the error and give the correct combined discount.
Read the worked answer
Adding percentages is wrong because the 10% applies to the already-reduced price, not the original. The correct combined multiplier is 0.5 × 0.9 = 0.45, meaning the customer pays 45% of full price - a 55% discount, not 60%. On $27.00: 0.45 × $27.00 = $12.15 paid, or $14.85 off. The add-the-percentages shortcut overstates the discount by 5 points and would undercharge customers if used at the register.