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?

$\$782$

$\$986$

$\$1158$

$\$1219$

$\$1449$

Answer: A

Concepts:Diophantine Equation prime factorization bounding to limit cases

Difficulty rating: 1490

Solution:

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.

2001 AMC 10 Problem 14

Answer: A

Difficulty rating: 1490

Solution:

Problem 14 in Other Years