Showcase Ticket Plan Under Three Constraints
A student council is selling tickets to a spring showcase. Adult tickets cost $8 and student tickets cost $5. The auditorium seats 300 people. The council must raise at least $1,800 to cover costs, and the principal requires at least 120 student tickets so the showcase stays affordable. The council plans to sell 150 adult tickets and 140 student tickets. One member says that because 290 of the 300 seats would be filled, the money goal is certain to be met. Another member wants to know the greatest number of adult tickets the council could sell if every seat is filled and both conditions still hold.
1. Check the council's plan of 150 adult and 140 student tickets against all three constraints (capacity, revenue, student minimum). Then evaluate the member's claim that filling 290 seats guarantees the money goal.
Read the worked answer
Check all three constraints. Seats: 150 + 140 = 290 ≤ 300. Revenue: 150($8) + 140($5) = $1,200 + $700 = $1,900 ≥ $1,800. Student minimum: 140 ≥ 120. The plan satisfies every condition.
The reasoning behind the member's claim is still wrong. Seat count does not determine revenue: 290 student tickets would bring in 290($5) = $1,450, far below the goal. The conclusion is true for this particular mix, but 'the seats are nearly full' is not evidence for it. A correct argument would compute the revenue.
2. Find the greatest number of adult tickets the council could sell if all 300 seats are sold and both the revenue and student-minimum conditions hold. Show the constraints and state which one limits the answer.
Read the worked answer
With all 300 seats sold, a + s = 300. The principal's rule gives s ≥ 120, so a ≤ 180. Revenue requires 8a + 5s ≥ 1,800. Substituting s = 300 − a: 8a + 5(300 − a) ≥ 1,800, so 3a + 1,500 ≥ 1,800, and a ≥ 100. Adult tickets must therefore fall between 100 and 180. The maximum is 180 adult tickets, paired with 120 student tickets, for revenue of $1,440 + $600 = $2,040. The student minimum limits the top of the range; revenue sets the floor. Without the principal's rule, 300 adult tickets would be possible.
3. A volunteer proposes 210 adult and 90 student tickets. Evaluate the proposal and explain which single change would make it acceptable without changing the total of 300 tickets.
Read the worked answer
210 adult and 90 student tickets fill all 300 seats and raise 210($8) + 90($5) = $1,680 + $450 = $2,130, so capacity and revenue are fine. The student minimum fails: 90 is 30 short of 120.
Moving 30 tickets from adult to student gives 180 adult and 120 student. Revenue becomes $1,440 + $600 = $2,040, still above $1,800. Each ticket shifted costs $3 in revenue, and 30 shifts cost $90, which the $330 surplus easily absorbs. That single change makes the proposal acceptable; changing only the total number of tickets would not fix the student minimum.