The admission fee at a water park is $18.25 for children and $35.50 for adults. On a certain day, 320 people entered the park, and the admission fees collected totaled $7,961.75. How many children and how many adults were admitted?

Let's define the variables and set up the equations based on the given information.

Let:
- $ c $ be the number of children.
- $ a $ be the number of adults.

From the problem, we have two pieces of information:
1. The total number of people: $ c + a = 320 $
2. The total admission fees collected: $ 18.25c + 35.50a = 7,961.75 $

We now have a system of linear equations:
\[ 
\begin{cases} 
c + a = 320 \\
18.25c + 35.50a = 7,961.75 
\end{cases}
\]

### Solving the system of equations:

1. Solve the first equation for one of the variables:
\[ c + a = 320 \implies c = 320 - a \]

2. Substitute $ c = 320 - a $ into the second equation:
\[ 18.25(320 - a) + 35.50a = 7,961.75 \]

3. Simplify and solve for $ a $:
\[ 18.25 \times 320 - 18.25a + 35.50a = 7,961.75 \]
\[ 5,840 - 18.25a + 35.50a = 7,961.75 \]
\[ 5,840 + 17.25a = 7,961.75 \]
\[ 17.25a = 7,961.75 - 5,840 \]
\[ 17.25a = 2,121.75 \]
\[ a = \frac{2,121.75}{17.25} \]
\[ a = 123 \]

4. Substitute $ a = 123 $ back into the first equation to find $ c $:
\[ c + 123 = 320 \]
\[ c = 320 - 123 \]
\[ c = 197 \]

Thus, the number of children admitted was 197, and the number of adults admitted was 123.

### Verification:
- Total number of people: $ 197 + 123 = 320 $ (correct)
- Total admission fees: $ 18.25 \times 197 + 35.50 \times 123 = 3,595.25 + 4,366.50 = 7,961.75 $ (correct)

The solution is correct: 197 children and 123 adults were admitted.

Would you like further details or have any questions about this solution?

Here are 5 more questions you might want to ask:

1. How do you solve a quadratic equation by factoring?
2. What is the derivative of a trigonometric function?
3. Can you explain the Pythagorean Theorem?
4. How do you find the area under a curve using integration?
5. What are the properties of logarithms?

**Tip:** Always double-check your solution by substituting the values back into the original equations to verify accuracy.

Learn how to solve a system of linear equations to determine the number of children and adults admitted to a water park. This problem involves calculating admission fees based on given rates and total revenue collected. Suitable for students in Grades 9-12.

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