The Downtown Theater group sold 650 tickets to its latest production. They sold only regular seats and box seats. Regular seats cost $12 each and box seats cost $18 each. Altogether, the theater collected $8,550 in ticket sales. How many regular seats and how many box seats were sold?
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To solve this problem, we can use a system of equations. Let's define two variables:
Let's call the number of regular seats sold "R".
Let's call the number of box seats sold "B".
According to the problem, the theater sold 650 tickets in total. This means that the sum of the regular and box seats sold is equal to 650:
R + B = 650 (Equation 1)
We are also given the total amount collected from ticket sales, which is $8,550. The revenue from regular seats (at $12 per ticket) can be calculated by multiplying the number of regular seats (R) by the price per regular seat ($12). Similarly, the revenue from box seats (at $18 per ticket) can be calculated by multiplying the number of box seats (B) by the price per box seat ($18). The sum of these revenues should equal the total revenue of $8,550:
12R + 18B = 8550 (Equation 2)
Now we have a system of equations that needs to be solved simultaneously:
R + B = 650 (Equation 1)
12R + 18B = 8550 (Equation 2)
We can now solve this system of equations to find the values of R and B.
One approach is to use the substitution method. We can solve Equation 1 for R in terms of B:
R = 650 - B
Now we substitute this value of R into Equation 2:
12(650 - B) + 18B = 8550
Simplifying this equation:
7800 - 12B + 18B = 8550
6B = 750
B = 125
Now that we have the value of B (number of box seats sold), we can substitute it back into Equation 1 to solve for R:
R + 125 = 650
R = 525
Therefore, 525 regular seats and 125 box seats were sold.