2014 AMC 10A Problem 11

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

Compare the actual dollar discount, not the final price

Solve the inequalities where coupon

1

beats coupons

2

and

3

Let us analyze what these coupons do to an arbitrary price,

x.

Coupon

1

changes this price to

0.9x.

Coupon

2

changes the price to

x - 20.

Coupon

3

changes the price to

x - 0.18(x - 100) = 0.82x + 18.

We want

0.9x \lt x - 20

and

0.9x \lt 0.82x + 18.

Solving both gives us

200 \lt x \lt 225.

The only answer choice that works is

\$ 219.95.

Thus, C is the correct answer.

Problem 11 in other years