Let \( a \) be the number of adults and \( c \) be the number of children. We have the following two equations based on the problem:
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The total number of attendees: \[ a + c = 500 \]
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The total amount of money earned from admissions: \[ 10a + 5c = 3500 \]
Now, we can solve these equations simultaneously.
From the first equation, we can express \( c \) in terms of \( a \): \[ c = 500 - a \]
Now, substitute this expression for \( c \) into the second equation: \[ 10a + 5(500 - a) = 3500 \] Distributing the 5: \[ 10a + 2500 - 5a = 3500 \]
Combine like terms: \[ 5a + 2500 = 3500 \]
Subtract 2500 from both sides: \[ 5a = 1000 \]
Now, divide by 5 to solve for \( a \): \[ a = 200 \]
Now substitute \( a \) back into the equation for \( c \): \[ c = 500 - a = 500 - 200 = 300 \]
Thus, the solution is: \[ \text{200 adults and 300 children} \]
Final answer: 200 adults; 300 children