2001 AMC 10 Problem 14

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

Let

n

tickets sell at full price

p;

then

np+(140-n)\dfrac p2=2001

This gives

p(n+140)

=4002

=2\cdot3\cdot23\cdot29,

with

140\le n+140\le280

Let

n

full-price tickets sell at

p

dollars each. Then

np+(140-n)\dfrac p2=2001,

so

p(n+140)

=4002

=2\cdot3\cdot23\cdot29.

Since

140\le n+140\le280,

the only factor of

4002

in range is

174=2\cdot3\cdot29.

So

n+140=174,

giving

n=34

and

p=23.

The full-price tickets raise

34\cdot23=782

dollars.

Thus, the correct answer is A.

Problem 14 in other years