We need to solve the problem by setting up a system of equations based on the given information.

Let:

• $x$ be the number of adult tickets sold.
• $y$ be the number of student tickets sold.
• $z$ be the number of child tickets sold.

The problem provides the following information:

1. Adult tickets cost $15, student tickets cost $10, and child tickets cost $8.
2. The total number of tickets sold is 325: $x + y + z = 325$
3. The total revenue is $3,700: $15x + 10y + 8z = 3700$
4. The number of student tickets sold is twice the number of adult tickets: $y = 2x$

Now we can use these three equations to solve for $x$, $y$, and $z$.

Step 1: Substitute $y = 2x$ into the first and second equations:

1. From $x + y + z = 325$, substitute $y = 2x$: $x + 2x + z = 325 \implies 3x + z = 325 \quad \text{(Equation 1)}$
2. From $15x + 10y + 8z = 3700$, substitute $y = 2x$: $15x + 10(2x) + 8z = 3700 \implies 15x + 20x + 8z = 3700 \implies 35x + 8z = 3700 \quad \text{(Equation 2)}$

Step 2: Solve the system of equations:

Now solve the system of two equations:

1. $3x + z = 325$
2. $35x + 8z = 3700$

Solve for $z$ from the first equation:

$z = 325 - 3x$ Substitute this into the second equation: $35x + 8(325 - 3x) = 3700$ $35x + 2600 - 24x = 3700$ $11x + 2600 = 3700$ $11x = 1100$ $x = 100$

Step 3: Solve for $y$ and $z$:

• $y = 2x = 2(100) = 200$
• $z = 325 - 3x = 325 - 3(100) = 25$

Final Answer:

• Number of adult tickets sold: $x = 100$
• Number of student tickets sold: $y = 200$
• Number of child tickets sold: $z = 25$

The department sold 100 adult tickets, 200 student tickets, and 25 child tickets.

Would you like more details or have any further questions?

Here are some related questions to think about:

1. What if the price of student tickets was changed, how would it affect the result?
2. How can we verify the total revenue calculation for different ticket prices?
3. What strategies could the theater department use to optimize ticket sales?
4. How would changes in the total number of tickets sold impact the system of equations?
5. What if the number of student tickets sold was not exactly double the adult tickets—how would the equations change?

Tip: Always double-check the units (such as dollars or number of tickets) when solving word problems!