https://file.aiquickdraw.com/mathbot/file/1727760086103zaj9rz3m.jpg

The problem described in the image involves maximizing the amount of money raised under specific constraints. Here’s the breakdown:

- **Adults pay $2.00** for admission.
- **Students pay $1.00** for admission.
- **The total number of attendees cannot exceed 180 people.**
- **For every 2 adults, there must be at least 1 student.**

Let:
- $ a $ be the number of adults attending.
- $ s $ be the number of students attending.

### Constraints:
1. The total number of people must not exceed 180:
   \[
   a + s \leq 180
   \]
2. There must be at least 1 student for every 2 adults:
   \[
   a \leq 2s
   \]

### Objective:
Maximize the total amount of money raised, which is:
\[
\text{Money} = 2a + s
\]
subject to the constraints $ a + s \leq 180 $ and $ a \leq 2s $.

### Solving the system:
1. From the second constraint, $ a \leq 2s $, we know that for maximum money, we want as many adults as possible (because adults pay more). So we can set $ a = 2s $ to maximize the number of adults.
   
2. Substitute $ a = 2s $ into the first constraint $ a + s \leq 180 $:
   \[
   2s + s \leq 180
   \]
   \[
   3s \leq 180
   \]
   \[
   s \leq 60
   \]

   So, the maximum number of students is $ s = 60 $. Using $ a = 2s $, the corresponding number of adults is:
   \[
   a = 2 \times 60 = 120
   \]

### Conclusion:
To raise the maximum amount of money, **120 adults** and **60 students** should attend.

Would you like more details or have any questions? 

Here are some related questions you can explore:
1. How would the solution change if the admission prices were different for adults and students?
2. What if the constraint on adults and students changed to a 1:1 ratio?
3. What is the total amount of money raised in this scenario?
4. How could you model this as a linear programming problem?
5. How would the maximum change if there was a fixed donation for each attendee?

**Tip:** Always check that your solution satisfies all constraints in the problem to ensure correctness.

A theater is presenting a program on drinking and driving for students and their parents or other responsible adults. The proceeds will be donated to a local alcohol information center. Admission is $2.00 for adults and $1.00 for students. The theater can hold no more than 180 people, and for every two adults, there must be at least one student. How many adults and students should attend to raise the maximum amount of money?

Learn how to determine the maximum amount of money raised in a theater event with attendance constraints. This problem involves linear equations, inequalities, and optimization, suitable for Grades 8-10.

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