The Discount That Doubled Back
A student in a middle-level math class computes the price of a $60 jacket marked down 25%, then marked down an additional 20% on the clearance rack. The student writes: "25% + 20% = 45% off, so 60 × 0.55 = $33." The teacher's answer key says $36. When asked to explain, the student insists the answer is $33 because "both discounts are taken off the price." The teacher wants to use this error as a teachable moment about percent change and successive percent operations rather than simply correcting the arithmetic. The class has already worked with percent bars and double number lines but has not yet formally studied compound percent change.
1. Explain the mathematical reasoning behind the correct answer, and identify precisely what misconception produced the student's $33.
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The correct answer is $36. The first markdown is 25% of $60 = $15, so the price becomes $45. The second markdown is 20% of $45 (the new price), not 20% of $60. That is $9, so the final price is $45 − $9 = $36.
The student's $33 comes from adding the two rates (25% + 20% = 45%) and applying 45% to the original $60. This is only valid when both rates are applied to the same base. Because the second discount applies to a smaller base ($45), the two discounts cannot be combined additively.
General check: successive percent changes multiply, not add. 0.75 × 0.80 = 0.60, so the final price is 60% of $60 = $36. The additive error overstates the discount by $3.
2. Describe a concrete representation or sequence of questions you would use to help the student see why the discounts cannot simply be added, and state what evidence would show the student has revised the misconception.
Read the worked answer
A double number line or percent-bar for the original $60, with a second bar for the $45 intermediate price, would let the student physically mark 25% on the first bar and then 20% on the second bar. The student can compare the length of 20% of $60 (which is $12) with 20% of $45 (which is $9) and see the difference.
A useful sequence of questions: (1) "After the first markdown, what is the price?" (2) "The clearance tag says 20% off - 20% of what?" (3) "If both discounts were 50%, would the jacket be free?" (This last question is a powerful counterexample: two successive 50% discounts give 25% of the original, not 0%.)
Evidence of revision: the student correctly computes successive discounts on a new problem (e.g., 10% then 10% off $200 gives $162, not $160), and explains in their own words that the second percent is taken from a smaller base. A transfer item where the student predicts the final price before computing, and justifies it with a percent bar, is stronger evidence than a single corrected answer.