Let's solve this problem step-by-step using algebra.

Step 1: Define the variables

Let:

• $C$ be the number of children.
• $S$ be the number of students.
• $A$ be the number of adults.

Step 2: Write the equations based on the problem conditions

1. Total number of attendees equation: $C + S + A = 303$

2. Relationship between the number of children and adults: $A = \frac{1}{2}C$

3. Total ticket sales equation: $5C + 7S + 12A = 2208$

Step 3: Substitute the relationship $A = \frac{1}{2}C$ into the equations

Substituting $A = \frac{1}{2}C$ into the total number of attendees equation: $C + S + \frac{1}{2}C = 303$ Multiplying the entire equation by 2 to clear the fraction: $2C + 2S + C = 606$ $3C + 2S = 606 \quad \text{(Equation 1)}$

Substituting $A = \frac{1}{2}C$ into the ticket sales equation: $5C + 7S + 12\left(\frac{1}{2}C\right) = 2208$ $5C + 7S + 6C = 2208$ $11C + 7S = 2208 \quad \text{(Equation 2)}$

Step 4: Solve the system of equations

We now have the following system of equations:

1. $3C + 2S = 606$
2. $11C + 7S = 2208$

Let's solve these equations using elimination or substitution. I'll proceed with elimination for clarity.

Multiply Equation 1 by 7 to align with the second equation: $21C + 14S = 4242$

Multiply Equation 2 by 2 to align with the first equation: $22C + 14S = 4416$

Subtract the first modified equation from the second: $(22C + 14S) - (21C + 14S) = 4416 - 4242$ $C = 174$

Step 5: Substitute $C = 174$ back into Equation 1

$3(174) + 2S = 606$ $522 + 2S = 606$ $2S = 84$ $S = 42$

Step 6: Find the number of adults using $A = \frac{1}{2}C$

$A = \frac{1}{2}(174) = 87$

Final Answer

• Number of children (C): 174
• Number of students (S): 42
• Number of adults (A): 87

Would you like more details on any of these steps, or do you have any questions?

Here are some related questions to expand on this problem:

1. What would be the new total ticket sales if the price for adults was increased to $15?
2. How would the distribution change if the total seating capacity increased to 350?
3. What happens to the number of students if the total number of adults becomes one-third of the children?
4. If the theater gave a discount of $2 for each category, how much would the total ticket sales be?
5. Can this problem be solved using matrix methods or a different algebraic approach?

Tip: In word problems, clearly defining variables and writing equations step-by-step makes it easier to organize information and find solutions.