Question

The ticket office makes $5 profit on each evening ticket sold and $3 on each matinee ticket sold. The ticket office wants to sell at least 50 evening tickets per week and at least 40 matinee tickets per week. Its goal is to earn at least $250 profit in sales per week. Show and describe all possible combinations of evening tickets and matinee tickets that the ticket office needs to sell to meet the goal. List two possible combinations.

Answer 1

Let E = number of evening tickets, M = number of matinee tickets.

Inequalities:
- E ≥ 50 (at least 50 evening tickets)
- M ≥ 40 (at least 40 matinee tickets)
- 5E + 3M ≥ 250 (profit requirement)

Check the minimums: 5(50) + 3(40) = 250 + 120 = 370 ≥ 250. Thus any pair (E, M) with E ≥ 50 and M ≥ 40 already gives at least $250 profit. So the feasible combinations are all (integer) pairs with
E ≥ 50 and M ≥ 40.

Two example combinations:
- (E, M) = (50, 40) → profit = 5·50 + 3·40 = $370
- (E, M) = (60, 45) → profit = 5·60 + 3·45 = $420

(Any other pair with E ≥ 50 and M ≥ 40 also works.)