imc

Question

A shopper plans to purchase an item that has a listed price greater than 100 and can use any one of the three coupons. Coupon A gives 15% off the listed price, Coupon B gives 30 off the listed price, and Coupon C gives 25% off the amount by which the listed price exceeds 100. Let x and y be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is y - x?

Accepted Answer

Let the listed price be (100 + p), where p > 0 Coupon A saves us: 0.15(100+p) = (0.15p + 15) Coupon B saves us: 30 Coupon C saves us: 0.25p Now, the condition is that A has to be greater than or equal to either B or C which gives us the following inequalities: A B 0.15p + 15 30 p 100 A C 0.15p + 15 0.25p p 150 We see here that the greatest possible value for p is 150, thus y = 100 + 150 = 250 and the smallest value for p is 100 so x = 100 + 100 = 200. The difference between y and x is y - x = 250 - 200 = 50