To find how many adults and how many children need to attend for the PTA to reach their goal of $3,500, we can set up a system of equations.
Let:
- \( a \) = the number of adults
- \( c \) = the number of children
From the problem, we have the following information:
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The total number of attendees (adults + children) is 500: \[ a + c = 500 \quad \text{(1)} \]
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The total revenue from admissions should be $3,500. The charge is $10 per adult and $5 per child: \[ 10a + 5c = 3500 \quad \text{(2)} \]
Now we can solve this system of equations.
First, we can simplify equation (2) by dividing everything by 5: \[ 2a + c = 700 \quad \text{(3)} \]
Now we can use equations (1) and (3) to eliminate \( c \). From equation (1), we can express \( c \) in terms of \( a \): \[ c = 500 - a \]
Next, we substitute this expression for \( c \) into equation (3): \[ 2a + (500 - a) = 700 \]
Simplifying that gives: \[ 2a + 500 - a = 700 \] \[ a + 500 = 700 \] \[ a = 700 - 500 \] \[ a = 200 \]
Now that we have the number of adults, we can find \( c \): \[ c = 500 - a = 500 - 200 = 300 \]
Thus, the solution is:
- Adults: 200
- Children: 300
Final Answer: 200 adults; 300 children