2001 AMC 10 Problem 14

Problem 14 of 25IntermediateAlgebra Number Theory

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?

Show solution

Solution

Let

n

full-price tickets sell at

p

dollars each. Then

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

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.

n+140=174,

giving

n=34

and

p=23.

The full-price tickets raise

34\cdot23=782

dollars. Thus, the correct answer is A.