Solving Math Word Problems: Movie Ticket Sales Analysis

This article takes a deep dive into a common math word problem involving the sale of movie tickets at different prices for adults and students. By applying basic algebra and logical reasoning, we can solve the puzzle and determine how many tickets of each type were sold. Learn how to systematically approach such problems and the techniques employed to find the solution.

Problem Details

The problem context is a common one: movie tickets were sold at $4.00 per adult and $2.50 per student. If 240 tickets in total were sold for a total revenue of $765.00, we aim to determine the number of student tickets and adult tickets sold.

Method 1: System of Equations

Step-by-Step Solution

To solve this problem, we start by assuming variables and creating equations based on the given data.

x represents the number of adult tickets sold. y represents the number of student tickets sold.

The first equation is based on the total number of tickets sold:[x y 240]

The second equation is based on the total revenue from the ticket sales:[4x 2.5y 765]

To solve these equations, we can use substitution or elimination methods. Let's use the substitution method for simplicity.

From the first equation, solve for x: [x 240 - y]

Substitute this expression for x into the second equation:[4(240 - y) 2.5y 765][960 - 4y 2.5y 765][960 - 1.5y 765][1.5y 195][y 130]

Now substitute y 130 back into the first equation to find x:[x 240 - 130][x 110]

Therefore, 110 adult tickets and 130 student tickets were sold.

Method 2: Logical Reasoning and Elimination

Step-by-Step Solution

Alternatively, we can use logical reasoning with some basic arithmetic:

If all tickets were sold to students, the total revenue would be: [2.5 times 240 600]

The difference between the actual revenue and the theoretical student revenue is:[765 - 600 165]

The difference in price between an adult ticket and a student ticket is:[4 - 2.5 1.5]

The number of adult tickets sold is:[165 / 1.5 110]

Thus, the number of student tickets is:[240 - 110 130]

Therefore, we confirm the same solution – 110 adult tickets and 130 student tickets.

General Solution Approach

This problem can be generalized into a system of linear equations. Consider a more complex scenario with adjusted prices and numbers of tickets sold. For example:

4x 2.5y 642.50 (Total revenue) x y 173 (Total number of tickets sold)

We can solve it by substitution or elimination. Here’s how:

Solve for x in terms of y from the second equation: [x 173 - y] Substitute into the first equation: [4(173 - y) 2.5y 642.50] [692 - 4y 2.5y 642.50] [692 - 1.5y 642.50] [1.5y 49.5] [y 33] Substitute y back into the first equation to find x: [x 173 - 33] [x 140]

Hence, 33 student tickets and 140 adult tickets were sold.

Conclusion

Math word problems like this can be solved using a variety of methods, including system of equations and logical reasoning. The key is to translate the problem into mathematical equations and solve them step-by-step. Understanding these techniques can help in tackling a wide range of real-world problems involving sales, budgeting, and financial planning.