2001 AMC 12 Problem 7

Question

A charity sells

140

benefit tickets for a total of

\$2001.

Some tickets sell for full price (a whole dollar amount), and the rest sell for half price. How much money is raised by the full-price tickets?

\$782

\$986

\$1158

\$1219

\$1449

With

n

full-price tickets at price

p,

the total is

np + (140 - n)\dfrac{p}{2} = 2001

This becomes

p(n + 140)

= 4002

= 2 \cdot 3 \cdot 23 \cdot 29,

and

n + 140

lies between

140

and

280

Let

n

tickets sell at full price

p

dollars. Then

np + (140 - n)\dfrac{p}{2} = 2001,

so

p(n + 140)

= 4002

= 2 \cdot 3 \cdot 23 \cdot 29.

Since

0 \le n \le 140,

we need a factor of

4002

with

140 \le n + 140 \le 280.

The only such factor is

174 = 2 \cdot 3 \cdot 29,

giving

n = 34

and

p = 23.

The full-price tickets raise

34 \cdot 23 = 782

dollars.

Thus, the correct answer is A.

Problem 7 in other years