Problem 11 · 2005 AMC 8
Easy
Fractions, Decimals & Percents
commutative-multiplication
The sales tax rate in Bergville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rings up $90.00, subtracts 20% of the price, then adds 6% of the discounted price for sales tax. What is Jack's total minus Jill's total?
Show answer
Answer: C — $0.
Show hints
Hint 1 of 2
Resist computing the two bills. Instead, write each as 90 times some multipliers — 'add 6%' is ×1.06, 'take 20% off' is ×0.80. Then just compare.
Still stuck? Show hint 2 →
Hint 2 of 2
Both clerks multiply by the same three numbers, only in a different order. What do you know about how order affects a product?
Show solution
Approach: see them as the same product reordered
- Translate each step into a multiplier: 'add 6% tax' is ×1.06, 'discount 20%' is ×0.80.
- Jack does 90 · 1.06 · 0.80; Jill does 90 · 0.80 · 1.06. Same three factors, swapped order.
- Multiplication doesn't care about order, so the totals are identical — the difference is $0.
- Why this transfers: stacked percentage changes are just multipliers, and multipliers commute. 'Discount then tax' always equals 'tax then discount' — recognizing this saves you from grinding out two dollar amounts (and from the ±$1.06 traps).
Mark:
· log in to save