Let's start by assigning variables to the unknowns:
Let x be the number of patron tickets sold.
Let y be the number of sponsor tickets sold.
Let z be the number of donor tickets sold.
We know that the number of donor tickets is 24 more than the sum of the patron and sponsor tickets combined, so we can write the equation z = x + y + 24.
We also know that the total number of tickets sold is 326, so we can write the equation x + y + z = 326.
Finally, we know that the total receipts from ticket sales is $1432.50, so we can write the equation 10x + 5y + 2.5z = 1432.50.
We now have a system of three equations with three variables:
z = x + y + 24,
x + y + z = 326,
10x + 5y + 2.5z = 1432.50.
We can solve this system using substitution or elimination. Let's use substitution:
First, we can substitute the expression for z from the first equation into the second equation:
x + y + (x + y + 24) = 326,
2x + 2y = 302.
Next, we can substitute the expression for z from the first equation into the third equation:
10x + 5y + 2.5(x + y + 24) = 1432.50,
10x + 5y + 2.5x + 2.5y + 60 = 1432.50,
12.5x + 7.5y = 1372.50.
Now we have a system of two equations with two variables:
2x + 2y = 302,
12.5x + 7.5y = 1372.50.
We can solve this system by multiplying the first equation by 6.25 to make the coefficients of y the same in both equations:
12.5x + 12.5y = 1887.50,
12.5x + 7.5y = 1372.50.
Now we can subtract the second equation from the first to eliminate x:
12.5x + 12.5y - (12.5x + 7.5y) = 1887.50 - 1372.50,
5y = 515,
y = 103.
Substituting this value of y back into the first equation, we can solve for x:
2x + 2(103) = 302,
2x + 206 = 302,
2x = 96,
x = 48.
Finally, substituting the values of x and y back into the equation for z, we can solve for z:
z = x + y + 24,
z = 48 + 103 + 24,
z = 175.
Therefore, the number of patron tickets sold is 48, the number of sponsor tickets sold is 103, and the number of donor tickets sold is 175.