A 3500-seat theater sells tickets for $75 and $110. Each night the theater's expenses total $245,000. When all 3500 seats sell, the owners want ticket revenues to cover expenses plus earn a profit of 25% of expenses. How many tickets of each price should be sold to achieve this?
X Tickets @ $75 each.
(3500-X) Tickets @ $110 each.
75x + 110(3500-x) = 245,000+0.25*245,000
75x + 385,000-110x = 306,250
-35x = 306,250-385,000 = -78,750
X = 2250 Seats @ $75 each.
3500-x=3500-2250 = 1250 seats @ @110each
Well, that's quite a big theater you've got there! Let's do some math and find out how many tickets of each price should be sold to achieve the desired profit.
Let's assume x represents the number of tickets sold at $75, and y represents the number of tickets sold at $110.
The total revenue from ticket sales would then be: 75x + 110y
According to the problem, this revenue needs to cover the expenses of $245,000 plus earn a profit of 25% of expenses. So the revenue should be 125% of the expenses, which can be expressed as: 1.25 * 245,000.
Putting it all together, we have the equation: 75x + 110y = 1.25 * 245,000
Now, since we know there are 3500 seats available, we can add another equation to represent that: x + y = 3500
Now we have a system of two equations. Solving them will give us the number of tickets of each price that should be sold.
However, since I'm more of a clown than a mathematician, I'll leave the solving part to you! Enjoy the number-crunching circus, my friend!
Let's assume the number of tickets sold for $75 is x and the number of tickets sold for $110 is y.
Based on the given information, we know that the total number of tickets sold should be 3500. So we have the equation:
x + y = 3500 ---(1)
The ticket revenue can be calculated by multiplying the number of tickets sold by their respective prices. So we have the equation:
75x + 110y = total revenue ---(2)
The owners want the ticket revenue to cover the expenses plus earn a profit of 25% of the expenses. So the total revenue should be equal to the expenses plus 25% of the expenses. Mathematically, this can be expressed as:
total revenue = expenses + 0.25 * expenses
total revenue = 1.25 * expenses
Given that the expenses are $245,000, we have:
total revenue = 1.25 * 245000
total revenue = $306,250
Substituting this value in equation (2), we get:
75x + 110y = 306250 ---(3)
Now we can solve equations (1) and (3) to find the values of x and y.
First, multiply equation (1) by 75 to eliminate x:
75x + 75y = 75 * 3500
75x + 75y = 262500 ---(4)
Subtract equation (4) from equation (3) to eliminate x:
(75x + 110y) - (75x + 75y) = 306250 - 262500
35y = 43750
Divide both sides by 35:
y = 43750 / 35
y = 1250
Substitute this value of y back into equation (1) to solve for x:
x + 1250 = 3500
x = 3500 - 1250
x = 2250
So, to achieve the desired result, 2250 tickets priced at $75 and 1250 tickets priced at $110 should be sold.
To solve this problem, we need to set up a system of linear equations to represent the given information.
Let's assume that x represents the number of $75 tickets sold and y represents the number of $110 tickets sold.
We know that there are a total of 3500 seats available, so we can write the equation:
x + y = 3500 (Equation 1)
We also know that the theater's expenses total $245,000 and the owners want ticket revenues to cover expenses plus earn a profit of 25% of expenses. So, the total revenue generated from ticket sales should be 125% of the expenses:
1.25 * 245000 = 306250 (Equation 2)
To determine how to express the total revenue, we need to consider the ticket prices. Each $75 ticket sold would generate $75 in revenue, and each $110 ticket sold would generate $110 in revenue. Therefore, the total revenue can be calculated as:
75x + 110y (Equation 3)
Since the total revenue should cover expenses plus earn a profit, we can equate Equations 2 and 3:
75x + 110y = 306250 (Equation 4)
Now we have a system of two equations (Equations 1 and 4) that can be solved simultaneously to find the values of x and y.
Using Equation 1, we can express x in terms of y:
x = 3500 - y
Substituting this expression into Equation 4, we get:
75(3500 - y) + 110y = 306250
262500 - 75y + 110y = 306250
35y = 43750
y = 1250
Substituting the value of y back into Equation 1:
x = 3500 - 1250
x = 2250
Therefore, to achieve the desired profit, 2250 tickets priced at $75 and 1250 tickets priced at $110 should be sold.