Successive Discounts: Why 20% + 10% Is Not 30%
A student is working on a Quantitative Reasoning comparison item. The problem states: 'A store offers two discount plans on a $200 item. Plan A: 20% off the original price, then an additional 10% off the reduced price. Plan B: 30% off the original price.' Quantity A is the final price under Plan A. Quantity B is the final price under Plan B. The student reasons that 20% + 10% = 30%, so the two plans must produce the same final price, and selects 'the two quantities are equal.' The student has seen similar problems where successive percentages were added directly and wants to know whether that shortcut is ever valid. The section allows a calculator, but the student prefers mental math. The student has 90 seconds to decide whether to change the answer and needs a clear method for comparing successive discounts.
1. Calculate Quantity A and Quantity B, show each step, and determine the correct comparison. Explain why the student’s addition shortcut fails.
Read the worked answer
Plan A: 20% off $200 means the first reduced price is $200 × 0.80 = $160. The additional 10% is taken off $160, not off $200: $160 × 0.10 = $16 discount, so final price = $160 − $16 = $144. Equivalently, $160 × 0.90 = $144. Plan B: 30% off $200 = $200 × 0.70 = $140. Quantity A = $144; Quantity B = $140. Since $144 > $140, Quantity A is greater. The student's addition shortcut fails because the second discount in Plan A applies to a smaller base ($160), not the original $200. Adding 20% + 10% treats the 10% as if it were 10% of $200 ($20), but it is only 10% of $160 ($16). The total effective discount under Plan A is $56/$200 = 28%, not 30%.
2. The student asks whether adding successive percentages is ever valid. Give a rule and a counterexample using the same numbers but a different order (10% then 20%).
Read the worked answer
Adding successive percentages is not valid for multiplicative changes unless one of the percentages is 0%. The order of successive percentage discounts does not matter for the final price when both are applied to the running total: 10% off then 20% off gives $200 × 0.90 = $180, then $180 × 0.80 = $144-same as 20% then 10%. But neither equals a single 30% discount ($140). A useful rule: a% off followed by b% off equals a single discount of 1 − (1−a)(1−b) on the original price. For 20% and 10%, 1 − 0.8×0.9 = 1 − 0.72 = 0.28, or 28%. Adding gives 30%, which is too high, so the student’s method would overstate the discount and understate the final price. In a comparison, that error could flip the answer.