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 $y - x?$

$50$

$60$

$75$

$80$

$100$

Solution:

Let $p$ be the price of the item. Then coupon A saves $.15p.$ Coupon B saves $\$ 30.$

Coupon C will save $.25(p - 100) = .25p - 25.$

We must have that $.15p \ge 30$ $p \geq 200$ and $.15p \ge .25p - 25$ $250 \geq p.$

This shows that $x = 200$ and $y = 250.$ Therefore $y - x = 50.$

Thus, A is the correct answer.

2010 AMC 10B Problem 11

Solution:

Problem 11 in Other Years