Solving Ticket Sales Problems with Algebra

Understanding how to solve equations involving ticket sales can be useful in many real-world scenarios, such as school plays or other events. In this article, we'll explore how to solve a classic ticket sales problem using algebra, and we'll provide step-by-step solutions for several variations of the problem. By the end, you will be able to solve similar problems with ease.

Problem Statement

For the school play, adult tickets cost $4.50 and student tickets cost $3.00. On a certain night, 325 tickets were sold, and the total revenue was $1,140. How many of each type of ticket were sold?

Step-by-Step Solution

Let's denote the number of adult tickets sold by A and the number of student tickets sold by C. We have two pieces of information that can be translated into two equations:

The total number of tickets sold is 325. The total revenue from ticket sales is $1,140.

We can write these as the following equations:

A C 325

4.5A 3C 1140

Step 1: Express C in terms of A

From the first equation, we can express C in terms of A:

C 325 - A

Step 2: Substitute C into the second equation

Substitute C 325 - A into the second equation:

4.5A 3(325 - A) 1140

Expand and simplify:

4.5A 975 - 3A 1140

1.5A 975 1140

1.5A 165

A 165 / 1.5

A 110

Step 3: Find the value of C

Now, substitute A 110 back into the first equation:

C 325 - 110

C 215

Conclusion

The number of adult tickets sold is 110, and the number of student tickets sold is 215.

Alternative Solutions

To provide further clarity, let's explore a couple of alternative solutions:

Solution 1

Let the number of adult tickets be x and the number of student tickets be y. We know:

x y 325

4.5x 3y 1140

From the first equation, express y in terms of x:

y 325 - x

Substitute this into the second equation:

4.5x 3(325 - x) 1140

4.5x 975 - 3x 1140

1.5x 165

x 110

y 325 - 110 215

The solution is the same as before: x 110 and y 215.

Solution 2

Let the number of adult tickets be A and the number of student tickets be 325 - A. We know:

Total receipts: 4A 3(325 - A) 1140

4A 975 - 3A 1140

A 975 1140

A 1140 - 975 165

A 110 and C 325 - 110 215

The solution remains the same.

Verification

To verify the solution, we can calculate the total revenue using the number of each type of ticket sold:

Total revenue from adult tickets: 110 × 4.50 $495.00

Total revenue from student tickets: 215 × 3.00 $645.00

Total revenue: $495.00 $645.00 $1,140.00

The solution is correct.

Summary

Solving ticket sales problems with algebra involves setting up and solving simultaneous equations. By following the steps outlined in this article, you can approach and solve similar problems with ease. Practice and familiarity with algebraic techniques will enhance your ability to handle real-world scenarios involving ticket sales and other financial calculations.