To solve this problem, we can use a system of equations based on the information given. Let:
- \( x \) be the number of adults attending the carnival.
- \( y \) be the number of children attending the carnival.
From the problem, we can set up the following equations:
-
The total number of people attending is 400: \[ x + y = 400 \]
-
The total revenue from admissions must be $2,500. The admission charges are $8 for adults and $3 for children: \[ 8x + 3y = 2500 \]
Now we can solve this system of equations. We can express \( y \) in terms of \( x \) using the first equation:
\[ y = 400 - x \]
Now, we can substitute this expression for \( y \) into the second equation:
\[ 8x + 3(400 - x) = 2500 \]
Expanding this gives:
\[ 8x + 1200 - 3x = 2500 \]
Combining like terms:
\[ 5x + 1200 = 2500 \]
Now, isolate \( x \):
\[ 5x = 2500 - 1200 \] \[ 5x = 1300 \] \[ x = \frac{1300}{5} = 260 \]
Now that we have \( x \), we can find \( y \):
\[ y = 400 - x = 400 - 260 = 140 \]
Thus, the number of adults that need to attend is 260 and the number of children that need to attend is 140.
To verify, we can check the revenue:
- Revenue from adults: \( 260 \times 8 = 2080 \)
- Revenue from children: \( 140 \times 3 = 420 \)
- Total revenue: \( 2080 + 420 = 2500 \)
The calculations are correct, so the answer is:
- 260 adults
- 140 children