Successive Discounts and Sales Tax
At a school store, a jacket has an original price of $80. The sign says '25% off, then take an additional 10% off the reduced price.' Sales tax of 8% is applied after both discounts. A student volunteer calculates the total for a customer as follows: '25% + 10% = 35% off, so the price is 0.65($80) = $52. Tax is 0.08($52) = $4.16, so the total is $56.16.' The customer pays, but later the manager checks the receipt and says the store undercharged. The student insists the arithmetic is correct because $52 plus $4.16 is $56.16. The manager wants the student to see that successive percent discounts do not add, and to use a single multiplier for the combined effect. The store also has a coupon that says 'additional 10% off the lowest marked price,' and the manager is considering a $60 maximum final total for a promotion.
1. Show why '25% off then 10% off' is not 35% off. Compute the correct pre-tax price, the sales tax, and the final total. How much did the student undercharge?
Read the worked answer
After 25% off, price = 0.75(80) = $60. The additional 10% applies to $60, not $80: 0.90(60) = $54. The combined multiplier is 0.75(0.90) = 0.675, so the total discount is 32.5%, not 35%. Tax = 0.08(54) = $4.32. Final total = $54 + $4.32 = $58.32. The student charged $56.16, so the undercharge is $2.16. Equivalently, the student used a pre-tax multiplier of 0.65 instead of 0.675; 80(0.675 - 0.65) = 80(0.025) = $2.00 pre-tax, which becomes $2.16 with tax.
2. Write a single decimal multiplier for the two discounts plus 8% tax. Use it to find the overall percent increase from the original price to the final total, and explain whether that percent is a discount or a markup.
Read the worked answer
The final-total multiplier is 0.75 × 0.90 × 1.08 = 0.729. That means the final total is 72.9% of the original price. The customer pays 27.1% less than the original price after tax, not 35% less. The combined discount before tax is 32.5%; tax adds 8% of the reduced price, so the net reduction is 27.1%. It is still a discount, not a markup, because 0.729 < 1.
3. For the promotion, the store wants the final total after both discounts and 8% tax to be at most $60. What is the maximum whole-dollar original price that works? Show the inequality and check your answer.
Read the worked answer
Let P be the original price. Final total = P(0.75)(0.90)(1.08) = 0.729P. Require 0.729P ≤ 60, so P ≤ 60 / 0.729, which is about 82.30. The largest whole-dollar price is $82. Check: 82(0.675) = $55.35 pre-tax; tax = 0.08(55.35) = $4.428; total = $59.778, which is at most $60. At $83, total = 83(0.729) = $60.507, over $60.