Math Problem Statement

5000 seat theater has tickets for sale at $24 and $40. How many tickets should be sold at each price for a sell out performance to generate a total revenue of $148,800?

Solution

Let the number of tickets sold at $24 be xx and the number of tickets sold at $40 be yy. We are given two key pieces of information:

  1. The total number of seats in the theater is 5000, so: x+y=5000x + y = 5000

  2. The total revenue is $148,800, so: 24x+40y=148,80024x + 40y = 148,800

Solving the system of equations:

  1. From the first equation, solve for yy: y=5000xy = 5000 - x

  2. Substitute this expression for yy into the second equation: 24x+40(5000x)=148,80024x + 40(5000 - x) = 148,800

  3. Expand the equation: 24x+200,00040x=148,80024x + 200,000 - 40x = 148,800

  4. Simplify: 16x+200,000=148,800-16x + 200,000 = 148,800

  5. Subtract 200,000 from both sides: 16x=51,200-16x = -51,200

  6. Divide by -16: x=3200x = 3200

So, 3200 tickets should be sold at $24.

  1. Now, substitute x=3200x = 3200 back into the equation for yy: y=50003200=1800y = 5000 - 3200 = 1800

Therefore, 1800 tickets should be sold at $40.

Final answer:

  • 3200 tickets should be sold at $24.
  • 1800 tickets should be sold at $40.

Would you like more details or have any further questions?

Here are some related questions:

  1. What is the total revenue if all tickets are sold at $24?
  2. How much additional revenue would result from increasing the $40 ticket price by 10%?
  3. If 100 fewer $24 tickets are sold, how many $40 tickets should be sold to maintain the same revenue?
  4. What percentage of the total seats are sold at the $40 price?
  5. How does the average ticket price change if 100 more $40 tickets are sold?

Tip: When solving word problems involving systems of equations, clearly define variables and equations based on the problem's constraints to simplify the process.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations

Formulas

x + y = 5000
24x + 40y = 148,800

Theorems

System of linear equations

Suitable Grade Level

Grades 9-12