Successive Percent Discounts: Adding Rates Taken from Different Bases
Mr. Alvarez gave his seventh graders this problem: A jacket is marked down 20%, and the sale price is then reduced by another 15%. Is the total discount 35%? Explain. Of 30 students, 18 answered yes, often writing 20 + 15 = 35. Four students computed 0.80 × 0.85 = 0.68 and said 32%. Two students insisted the answer depends on the original price. One student drew a bar and labeled the second discount on the original price. Mr. Alvarez has 25 minutes the next day and wants to use the disagreement to build understanding rather than simply state the rule. He also needs a quick way to check whether students can transfer the reasoning to a 25%-then-10% discount.
1. Using a $200 original price, show why 35% is not correct and identify the reasoning error behind the 18 responses.
Read the worked answer
At $200: the first discount is 20% of $200 = $40, so the sale price is $160. The second discount is 15% of $160 = $24, so the final price is $136. The total discount is $64, and $64/$200 = 0.32 = 32%, not 35%.
The error is adding rates that apply to different bases. The 20% is taken from the original price, but the 15% is taken from the reduced price. If the second 15% were taken from the original $200, it would be $30 and the final price would be $130, a true 35% discount. The two students who said it depends on the original price are partly right about the dollar amount: at $80 the total dollar discount is $25.60, not $64. But the percent discount remains 32% because both discounts are multiplicative: 0.80 × 0.85 = 0.68. The bar model showing 15% of the original price is the key visual error.
2. Describe a two-step instructional sequence, with a representation, that would help students see why successive percent changes multiply. Include a quick transfer check.
Read the worked answer
Sequence: (1) Draw a tape diagram of $200 in five equal $40 parts. Cross out one part for 20% off, leaving $160. Then draw 15% as 15 small units out of 100 on the $160 bar, or compute 0.15 × 160 = $24, leaving $136. Ask students to compare that final price with a single 32% discount: 0.68 × 200 = $136. (2) Have pairs test a second price, such as $80: 20% off gives $64; 15% of $64 is $9.60; final $54.40; total discount $25.60, which is 32% of $80. Then ask them to generalize: 0.80 × 0.85 = 0.68, so the combined multiplier is 0.68, a 32% discount. This builds the idea that successive percent changes multiply rather than add. Quick transfer check: 'A $120 item is discounted 25%, then 10%. What single percent discount is equivalent?' Expected: 0.75 × 0.90 = 0.675, so 32.5% off, final $81. Also ask which is better, 30% off or 20% then 10% (30% is better).