Question

Theater Revenues A movie theater charges \(\$ 11.00\) for adults, \(\$ 6.50\) for children, and \(\$ 9.00\) for senior citizens. One day the theater sold 405 tickets and collected \(\$ 3315\) in receipts. Twice as many children's tickets were sold as adult tickets. How many adults, children, and senior citizens went to the theater that day?

Short answer

110 adults, 220 children, and 75 senior citizens

Step-by-step solution

Define Variables

Let \(a\) represent the number of adult tickets sold, \(c\) represent the number of children's tickets sold, and \(s\) represent the number of senior citizen tickets sold.

Set Up Equations

We have three pieces of information that lead to three equations:1. The total number of tickets: \(a + c + s = 405\).2. The total money collected: \(11a + 6.5c + 9s = 3315\).3. Twice as many children's tickets as adult tickets: \(c = 2a\).

Substitute \(c\)

Substitute \(c = 2a\) into the first and second equations:1. \(a + 2a + s = 405\), simplified to \(3a + s = 405\).2. \(11a + 6.5(2a) + 9s = 3315\), simplified to \(11a + 13a + 9s = 3315\) or \(24a + 9s = 3315\).

Solve for \(s\)

From the first simplified equation: \(s = 405 - 3a\).Substitute \(s\) into the second simplified equation: \(24a + 9(405 - 3a) = 3315\).

Simplify and Solve for \(a\)

Expand and combine like terms:\[24a + 3645 - 27a = 3315\]Combine \(24a\) and \(-27a\):\[-3a + 3645 = 3315\]Isolate \(a\):\[-3a = -330\]\[a = 110\]

Find \(c\) and \(s\)

Use \(a = 110\) to find \(c\) and \(s\):\(c = 2a = 2(110) = 220\).\(s = 405 - 3a = 405 - 3(110) = 75\).

Verify the Solution

Check that the values satisfy all original equations:1. Number of tickets: \(a + c + s = 110 + 220 + 75 = 405\), which is correct.2. Total receipts: \(11(110) + 6.5(220) + 9(75) = 1210 + 1430 + 675 = 3315\), which is also correct.

Key concepts

linear equations

Linear equations are mathematical expressions that represent straight lines when graphed on a coordinate plane. They are typically written in the form of 𝑎𝑥 + 𝑏𝑦 = 𝑐, where 𝑎, 𝑏, and 𝑐 are constants, and 𝑥 and 𝑦 are variables. These equations are fundamental in algebra and can describe a range of real-world situations, like the ticket sales problem described. In this problem, we used three linear equations to represent:

substitution method

The substitution method is used to solve systems of linear equations by solving one of the equations for one variable, then substituting that expression into the other equation(s). This method simplifies the system and can be done in a few steps:
1. Solve one equation for one variable in terms of the other(s).
2. Substitute this expression into the other equation(s).
3. Solve the resulting equation for the remaining variable.
4. Substitute back to find the value of the original variable.
This method works well for systems where at least one equation is easily solved for one variable, as seen in the ticket sales problem where we solved c = 2a and used it in other equations.

solving equations

Solving equations involves finding the values of variables that make the equation true. These values are called the solution or solutions. To solve the equations in our problem, we followed these steps:
1. **Simplify the equations:** Combine like terms and use arithmetic operations to reduce the complexity of equations.
2. **Substitute known values:** Replace variables with their known values from other equations to simplify.
3. **Isolate variables:** Use inverse operations (like addition/subtraction or multiplication/division) to isolate the variable on one side of the equation.
This systematic approach ensures you find the correct solution accurately and efficiently.