2010 AMC 10B Problem 11

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

Write each coupon’s savings in terms of the listed price

Coupon A must beat both

\$30

and

0.25(p-100)

Let

p

be the price of the item. Then coupon A saves

0.15p.

Coupon B saves

\$ 30.

Coupon C will save

0.25(p - 100) = 0.25p - 25.

We must have that

0.15p \ge 30

p \geq 200

and

0.15p \ge 0.25p - 25

250 \geq p.

This shows that

x = 200

and

y = 250.

Therefore

y - x = 50.

Thus, A is the correct answer.

Problem 11 in other years