Question

A theater owner wants to divide a 3300 seat theater into three sections, with tickets costing $20, $90, and $120, depending on the section. He wants to have twice as many $20 tickets as the sum of the other tickets, and he wants to earn $155,000 from a full house. Find how many seats he should have in each section. What is the number of $20 seats is?

Answer 1

To solve the theatre owner’s problem, we set up a system of equations based on the given conditions, such as the total number of seats, the revenue goal, and the ratio between 20 tickets and other tickets, and then we solve the system to find the number of seats per section.

Step-by-step explanation:

Let's denote the number of 20 tickets as x, the number of 90 tickets as y, and the number of 120 tickets as z. Given that the theatre owner wants twice as many $20 tickets as the sum of the other tickets, we can write the first equation as:

x = 2(y + z).

The total number of seats is 3300, which gives us the second equation:

x + y + z = 3300.

To meet the revenue goal of 155,000, we get the third equation:

20x + 90y + 120z = 155,000.

Substituting the first equation into the other two, we can solve the system of equations using elimination or substitution methods to obtain the number of seats in each section:

1. Replace x with 2(y + z) in the second and third equations.
2. Solve the resulting two equations for y and z.
3. Substitute y and z back into x = 2(y + z) to find x.

Once the values of x, y, and z are found, we will know how many seats the theatre owner should allocate to each section and particularly how many 20 seats there are.