Percent Discount and Tax: Additive vs Multiplicative Reasoning
During a seventh-grade team meeting, teachers examine three responses to a store problem: a jacket regularly costs $80. The store advertises 20% off the regular price, and an 8% sales tax is applied to the discounted price. Student A writes 80 − 0.20(80) + 0.08(80) = $70.40. Student B writes 80(0.80)(1.08) = $69.12. Student C writes 80(1 − 0.20 + 0.08) = $70.40. The team wants to use the responses to plan a short reteach on percent models and the order of operations. A coupon offers an additional 10% off the discounted price, and the teachers need to know whether applying that coupon before or after the 8% tax changes the final cost. They agree the reteach must distinguish additive and multiplicative reasoning without simply giving students a formula to memorize.
1. Which student's arithmetic is correct, and what conceptual error is shown by each of the other two? Write a concise explanation suitable for the reteach.
Read the worked answer
Student B is correct. The discounted price is 0.80 × $80 = $64.00. Tax is 8% of that discounted price, so tax is 0.08 × $64 = $5.12, and the final cost is $64.00 + $5.12 = $69.12.
Student A applies the 8% tax to the original $80, not to the $64 discounted price. Student A's tax is 0.08 × $80 = $6.40, which is too large. Student C combines the 20% discount and 8% tax additively as 1 − 0.20 + 0.08 = 0.88. That treats both percents as if they were applied to the same base and as if they could be added. The correct multiplier is 0.80 × 1.08 = 0.864, so the final price is 86.4% of the original, not 88%.
2. The coupon is 10% off the discounted price. Compare the final cost if the coupon is applied before tax with the cost if it is applied after tax. Show calculations and explain whether the order matters.
Read the worked answer
Before tax: start with $80, take 20% off to get $64, take 10% off $64 to get $57.60, then add 8% tax: $57.60 × 1.08 = $62.208, which rounds to $62.21.
After tax: start with $80, take 20% off to get $64, add 8% tax to get $69.12, then take 10% off the after-tax amount: $69.12 × 0.90 = $62.208, which rounds to $62.21.
The order does not change the final cost because the calculation is a product: 80 × 0.80 × 0.90 × 1.08 in the first case and 80 × 0.80 × 1.08 × 0.90 in the second. Multiplication is commutative. Rounding at intermediate steps could move the result by a cent, but the underlying multiplicative relationship is the same. This would not be true if the coupon were defined as 10% of the original price rather than 10% off the current discounted price.
3. Design a tape diagram or number-line representation that would help students see why the correct multiplier is 0.80 × 1.08 rather than 1 − 0.20 + 0.08. Describe the parts and labels.
Read the worked answer
Draw a bar labeled $80 as the whole. Divide it into 5 equal parts of $16 each. Mark one part as the 20% discount and remove it, leaving four parts labeled $64. This shows that the discount leaves 80% of the original price.
Then draw a second bar of length $64. Since 8% is 8/100, divide this bar into 100 equal parts or use a proportional segment labeled 0.08 × $64 = $5.12. Add that segment to the $64 bar to get $69.12. The full process is 80 × 0.80 × 1.08 = 69.12, so the final amount is 86.4% of the original.
The additive expression 1 − 0.20 + 0.08 would be a single bar where 20% and 8% are both taken from the same $80 whole. That diagram would represent subtracting $16 and adding $6.40, not taking tax on the reduced $64 price.