Successive Percents and the Wrong Base
Ms. Alvarez's seventh-grade class is working a multi-step percent problem. A jacket is priced at $84.99, and a sign reads: "Take 30% off - then take an extra 15% off the reduced price." Sales tax of 7.5% is added after all discounts. Two students share answers. Devon says the total is $50.25, explaining that 30% + 15% = 45% off. Priya correctly gets $50.57 before tax but then computes the 7.5% tax on the original $84.99, giving $56.94. A third student, Marcus, asks whether applying the 15% first and the 30% second would change the answer. Ms. Alvarez has ten minutes before the period ends and wants to use the disagreement to teach something durable rather than simply announcing the right answer. The class has already studied the distributive property and moves fluently among fractions, decimals, and percents, but successive-percent problems are new.
1. Compute the correct final price and explain why "30% + 15% = 45% off" is not valid, using both a concrete counterexample and a general argument.
Read the worked answer
Sequential discounts: $84.99 × 0.70 = $59.49 (the reduced price); $59.49 × 0.85 = $50.57 (after the extra 15%); $50.57 × 1.075 = $54.36 final.
Counterexample with $100. Sequential: 100 × 0.70 × 0.85 = $59.50. Single 45% discount: 100 × 0.55 = $55.00. Same starting price, different results, so the two procedures are not equivalent.
General argument. Each discount multiplies by a factor: (1 − 0.30)(1 − 0.15) = 0.70 × 0.85 = 0.595, a 40.5% total reduction, not 45%. Adding the percents implicitly uses the factor 1 − (0.30 + 0.15) = 0.55, which is smaller. The reason is the base: the 15% is taken on $59.49 (about $8.92), not on $84.99 ($12.75). Percent sentences are multiplicative comparisons - "15% of what?" - and the second base has already shrunk.
Priya's error is a base error of the same type: tax is 7.5% of the amount actually paid, $50.57, giving $3.79, for $54.36. Taxing $84.99 overcharges by about $2.58. Both errors trace to fixing the wrong whole.
2. Marcus asks whether reversing the order of the two discounts changes the result. Answer him, justify with a property, and explain what the order actually does and does not affect.
Read the worked answer
Reversing the order gives the same final price, because the calculation is a product of factors and multiplication is commutative and associative: 84.99 × 0.85 × 0.70 = 84.99 × 0.70 × 0.85 = $50.57. Even the tax factor can be placed anywhere in the product without changing the result, since 84.99 × 0.70 × 0.85 × 1.075 is one product.
What order does change is the intermediate dollar amounts, and therefore how the situation reads to a customer: after 30% first, the jacket shows $59.49 and the 15% saves $8.92; after 15% first, it shows $72.24 and the 30% saves $21.67. The savings attributed to each discount differ, but they cancel to the same total.
What order does not rescue is a wrong factor. Devon's 0.55 is wrong regardless of sequencing because it changes the base structure, not the order of operations. One genuine practical exception: rounding after each step. If the register rounds $59.493 to $59.49 before applying 15%, the result shifts by a cent or two, so the order of rounding can matter even though the order of exact multiplication cannot.
3. What instructional move should Ms. Alvarez make in the remaining ten minutes, and which representation best exposes the error? Justify the choice against the alternative of simply correcting the answer.
Read the worked answer
She should have students test both procedures on a friendly number rather than tell them the rule. Pose: "A $100 jacket, 30% off then 15% off - what do you pay? Now what would 45% off give?" The two results ($59.50 versus $55.00) create a surprise that students must resolve, which keeps the cognitive demand on them.
Representation: a bar or tape diagram. Draw one bar for $84.99, shade 70% of it as the reduced price, then draw a second, shorter bar representing that 70% portion and shade 85% of it. The diagram makes visible what a symbolic factor hides: the second percent is cut from the already-shortened bar, so the two shaded regions cannot be added as 45%. Pair it with the expression 84.99(0.70)(0.85)(1.075) so the visual maps onto the algebra.
For Priya, one question suffices: "7.5% of what?" Have her locate the bar that represents the price actually paid. Simply announcing "it's 40.5%, not 45%" would leave the additive misconception intact; the class would memorize a correction without repairing the base reasoning, and the same error would resurface with tips, markups, and percent change.