Problem 11 · 2003 AMC 8
Medium
Fractions, Decimals & Percents
percent-multiplier
Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by 10%. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40 on Thursday?
Show answer
Answer: B — $39.60.
Show hints
Hint 1 of 2
Tempting to say up 10% then down 10% cancels — but the 10% cut comes off the new, larger price, so it removes more than the increase added.
Still stuck? Show hint 2 →
Hint 2 of 2
Turn each percent change into a multiplier and chain them: ×1.1 (up 10%) then ×0.9 (down 10%).
Show solution
Approach: turn each change into a multiplier and chain them
- Each percent change is a multiplier: a 10% raise is ×1.1, a 10% cut is ×0.9. Apply them in order: Friday 40 × 1.1 = 44, then Monday 44 × 0.9 = 39.60.
- The order doesn't even matter, since multipliers just combine: 1.1 × 0.9 = 0.99, so the price ends at 40 × 0.99 = 39.60 — below the start, not back to it.
- Worth keeping: +x% then −x% always lands a little low, because 0.99 < 1 (in general (1+r)(1−r) = 1 − r²). Percent changes multiply, they don't add — so they never simply cancel.
Mark:
· log in to save