Solving a Ticket Sales Puzzle: An Educational and Engagement-Focused Approach

Have you ever found yourself solving number puzzles while attending a school concert? Let's explore a real-world scenario involving ticket sales and see how we can apply mathematical reasoning to solve this conundrum.

The Scenario

A local school hosted a concert, selling a total of 120 tickets. These tickets came in two categories: children's tickets priced at $5 each and adult tickets priced at $8 each. The total revenue collected from the ticket sales was $708. Now, the question at hand is: How many children attended the concert?

A Step-by-Step Solution

Setting Up the Equations

Let's denote the number of children's tickets sold as x and the number of adult tickets sold as y.

We have two key pieces of information that translate into the following equations:

Total number of tickets sold: Total revenue collected:

Mathematically, these can be written as:

x y 120 5x 8y 708

Solving for One Variable

Our strategy will be to solve these equations step by step. Let's start with the first equation to express y in terms of x.

y 120 - x

Substituting into the Second Equation

Next, we substitute this expression for y into the second equation:

5x 8(120 - x) 708

Expanding and Simplifying

Now, let's expand and simplify the equation:

5x 960 - 8x 708

-3x 960 708

-3x 708 - 960

-3x -252

x 84

Finding the Number of Adults

Now that we have the value of x, let's find y:

y 120 - 84 36

So, the number of children who attended is 84.

Alternative Methods and Verification

Method 1: Maximum Revenue Scenario

Let's consider the scenario where all the tickets sold are to adults. The total cost would be:

143 adults * $8 $1,144

The difference between the ideal revenue and the actual revenue is:

$1,144 - $708 $436

Since children's tickets are $4 cheaper than adult tickets:

$436 / $4 109 children

This method yields a slightly different result, indicating an error or an assumption not met in the original problem.

Method 2: Substitution

Using a for the number of adults and s for the number of students:

a s 143

12a 8s 1368

Solving for s:

12(143 - s) 8s 1368

1716 - 12s 8s 1368

-4s -348

s 87

Again, substituting back into the equation for s:

s 87

Thus, 87 students attended the concert.

Finally, verifying our solution confirms that 84 children attended the concert.

Conclusion

Through careful mathematical analysis, we can solve real-world problems like ticket sales. The method of solving linear equations not only helps in understanding the distribution but also in checking the accuracy of the solution. This problem-solving approach can be applied to various scenarios, enhancing both educational and practical skills.

Key Learning Points:

Understanding and setting up linear equations. Substitution and elimination methods in solving equations. Verification of solutions to ensure accuracy.