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2014 AMC 12A Problem 8

Problem 8 of 25EasierAlgebra

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?

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Solution

For a price P>100,P\gt100, the reductions are P10,\dfrac{P}{10}, 20,20, and 18100(P−100).\dfrac{18}{100}(P-100). Coupon 11 beats coupon 22 when P10>20,\dfrac{P}{10}\gt20, that is P>200.P\gt200. Coupon 11 beats coupon 33 when P10>18100(P−100),\dfrac{P}{10}\gt\dfrac{18}{100}(P-100), that is P<225.P\lt225. The only listed price in (200,225)(200,225) is $219.95.\$219.95. Thus, the correct answer is C.

More practice

Concepts: inequality · percentage

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