A charity sells 140 benefit tickets for a total of 2001. Some tickets

Average price per ticket \(= \frac{revenue}{quantity} = \frac{2001}{140}\) ≈ 14.3
Implication:
A full-price ticket must cost MORE than $14.30, while a half-price ticket must cost LESS than $14.30.

Total revenue = 2001 = 3*23*29.
Since the price of a full-price ticket must be an INTEGER value greater than 14.3, the prime-factorization above implies that a full-price ticket probably costs either $23 or $29, yielding the following options:
Full-price = 29, half-price = 14.5
Full-price = 23, half-price = 11.5
Since a half-price ticket must cost less than $14.30, only the green option is viable.

Let F = the number of full-price tickets and H = the number of half-price tickets.
Since the total revenue is $2001, we get:
23F + 11.5H = 2001
46F + 23H = 4002

Since a total of 140 tickets are sold, we get:
F + H = 140
23F + 23H = 3220

Subtracting the red equation from the blue equation, we get:
23F = 782
F = 34
Success!
The number of full-price tickets is an integer.

The equation in green indicates that the total revenue yielded by the $23 full-price tickets = 782.