Question

A customer who intends to purchase an appliance has three coupons, only one of which may be used:

Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$

Coupon 2: $\$20$ off the listed price if the listed price is at least $\$100$

Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$

For which of the following listed prices will coupon 1 offer a greater price reduction than either coupon 2 or coupon 3?

$\$179.95$

$\$199.95$

$\$219.95$

$\$239.95$

$\$259.95$

Solution:

For a price $P\gt100,$ the reductions are $\dfrac{P}{10},$ $20,$ and $\dfrac{18}{100}(P-100).$

Coupon 1 beats coupon 2 when $\dfrac{P}{10}\gt20,$ that is $P\gt200.$ Coupon 1 beats coupon 3 when $\dfrac{P}{10}\gt\dfrac{18}{100}(P-100),$ that is $P\lt225.$

The only listed price in $(200,225)$ is $\$219.95.$

Thus, the correct answer is C.

2014 AMC 12A Problem 8

Solution:

Problem 8 in Other Years