Fundraiser Ticket Mix and Profit Threshold
A community theater has 540 seats for a fundraiser. Adult tickets sell for $18 and student tickets for $11. The event has fixed costs of $4,200. Organizers want a profit of at least $3,600 after the event. They also promised that at least 200 of the tickets sold will be student tickets. All 540 seats are expected to sell. Ticket revenue is the only income. Let $a$ be adult tickets and $s$ be student tickets. The organizers need to know whether their ticket mix can satisfy both the profit goal and the student-ticket promise.
1. If the event sells out and takes in $7,620 in ticket revenue, write and solve a system of equations for the number of adult and student tickets sold.
Read the worked answer
The system is
$a + s = 540$
$18a + 11s = 7620$.
Substitute $s = 540 - a$ into the revenue equation:
$18a + 11(540 - a) = 7620$
$18a + 5940 - 11a = 7620$
$7a = 1680$
$a = 240$.
Then $s = 540 - 240 = 300$. The event sold 240 adult tickets and 300 student tickets. Check: $18(240) + 11(300) = 4320 + 3300 = 7620$.
2. Using the ticket counts from the first task, does the event meet the $3,600 profit goal? If fixed costs rise to $4,650, find the minimum number of adult tickets that must replace student tickets, keeping total sales at 540, to reach a $3,600 profit.
Read the worked answer
At 240 adult and 300 student tickets, revenue is $7,620, so profit is
$7620 - 4200 = 3420$,
which is $180 short of $3,600.
With fixed costs of $4,650, required revenue is $4,650 + 3,600 = 8,250. Keeping total sales at 540:
$18a + 11(540 - a) \ge 8250$
$7a + 5940 \ge 8250$
$7a \ge 2310$
$a \ge 330$.
She needs at least 330 adult tickets, which is 90 more than the original 240. That means replacing 90 student tickets with adult tickets, leaving 210 student tickets.
3. The student-ticket promise requires $s \ge 200$. Is the $3,600 profit still achievable after the cost increase? State the range of adult tickets that satisfies both the profit and student-ticket conditions.
Read the worked answer
From the profit condition, $a \ge 330$. From the student-ticket promise, $s \ge 200$, and since $a + s = 540$, this means $a \le 340$.
So the adult-ticket count must satisfy
$330 \le a \le 340$,
with student tickets between 200 and 210. This is feasible. At $a = 330$, revenue is $18(330) + 11(210) = 5940 + 2310 = 8250$, and profit is $8250 - 4650 = 3600$, exactly meeting the goal. At $a = 340$, profit is $3670$. So the goal is achievable, but only with a narrow ticket mix.