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2014 AMC 10A Problem 11

Problem 11 of 25IntermediateAlgebra

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1:1: 10%10\% off the listed price if the listed price is at least $50\$50 Coupon 2:2: $20\$ 20 off the listed price if the listed price is at least $100\$100 Coupon 3:3: 18%18\% off the amount by which the listed price exceeds $100\$100 For which of the following listed prices will coupon 11 offer a greater price reduction than either coupon 22 or coupon 3?3?

Answer choices

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Solution

Let us analyze what these coupons do to an arbitrary price, x.x. Coupon 11 changes this price to 0.9x.0.9x. Coupon 22 changes the price to x20.x - 20. Coupon 33 changes the price to x0.18(x100)=0.82x+18. x - 0.18(x - 100) = 0.82x + 18. We want 0.9x<x20 0.9x \lt x - 20 and 0.9x<0.82x+18. 0.9x \lt 0.82x + 18. Solving both gives us 200<x<225.200 \lt x \lt 225. The only answer choice that works is $219.95.\$ 219.95. Thus, C is the correct answer.

More practice

Concepts: inequality · percentage

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.