Let the number of children's tickets sold be \( c \) and the number of adult tickets sold be \( a \).
We have the following two equations based on the problem's conditions:
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The total number of tickets sold: \[ c + a = 450 \]
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The total revenue from ticket sales: \[ 2c + 5a = 1800 \]
We can solve these equations step by step.
First, from the first equation, we can express \( c \) in terms of \( a \): \[ c = 450 - a \]
Next, we substitute \( c \) in the second equation: \[ 2(450 - a) + 5a = 1800 \]
Expanding this, we get: \[ 900 - 2a + 5a = 1800 \]
Combining like terms: \[ 900 + 3a = 1800 \]
Now, we isolate \( a \): \[ 3a = 1800 - 900 \] \[ 3a = 900 \] \[ a = \frac{900}{3} = 300 \]
Thus, the number of adult tickets sold is \( \boxed{300} \).
We can also find the number of children's tickets sold for verification: \[ c = 450 - a = 450 - 300 = 150 \]
Now checking the revenue: \[ 2c + 5a = 2(150) + 5(300) = 300 + 1500 = 1800 \]
The calculations confirm that the solution is correct, and James sold \( \boxed{300} \) adult tickets.
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