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: 1370

Solution:

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.

2001 AMC 12 Problem 7

Answer: A

Difficulty rating: 1370

Solution:

Problem 7 in Other Years