Question

Ticket Sales You are the manager of a theater. On Saturday morning you are going over the ticket sales for Friday evening. A total of 740 tickets were sold. The tickets for adults and children sold for $$\$ 8.50$$ and $$\$ 4.00$$, respectively, and the total receipts for the performance were $$\$ 4688$$. However, your assistant manager did not record how many of each type of ticket were sold. From the information you have, can you determine how many of each type were sold? Explain your reasoning.

Short answer

After calculating the values from the equations, you will find that there were 356 adult tickets and 384 child tickets sold.

Step-by-step solution

Formulate the equations

Let's assume the number of adult tickets sold is \(x\) and the number of child tickets sold is \(y\). Therefore, according to the given condition, we can make two equations as follows: \(x + y = 740\) (since the total number of tickets sold was 740), and \(8.5x + 4y = 4688\) (the total money collected was $4688, with $8.50 per adult ticket and $4.00 per child ticket)

Solving the equations

Now we have two equations with two variables which can be solved. We'll use one approach to solve this kind of equations system: substitution or elimination. For this exercise, let's use the substitution method. We can isolate \(x\) in the first equation \(x = 740 - y\) and substitute \(x\) in the second equation with \(740 - y\), so it becomes \(8.5(740 - y) + 4y = 4688\)

Simplify and solve for y

Simplify the equation into: \(6290 - 8.5y + 4y = 4688\), resulting in an equation that has only one variable \(y\), \(4.5y = 1602\), then divide both sides by 4.5 to give: \(y = \frac{1602}{4.5}\) calculate the value of \(y\) to get the number of child tickets sold.

Substitute y in the first equation to find x

Substitute the value of \(y\) we found in the first equation, \(x = 740 - y\), to get the number of adult tickets.

Key concepts

Understanding the Substitution Method

The substitution method is a technique used in algebra to solve systems of equations where one variable is expressed in terms of the other and then substituted into the other equation. This is particularly useful when equations are not conveniently arranged for elimination.

In our ticket sales problem, we started with two equations: 1) The total ticket count \(x + y = 740\) and 2) The total revenue \(8.5x + 4y = 4688\). The first step is to express one variable in terms of the other from the total ticket count. We chose to solve for \(x\) which gives us \(x = 740 - y\).

This expression for \(x\) was then substituted into the revenue equation, replacing every instance of \(x\) with \(740 - y\). This single-variable equation could now be solved to find the exact number of child tickets sold (\(y\)). Once \(y\) was calculated, it was substituted back into the original expression for \(x\) to find the number of adult tickets sold.

Algebraic Problem-Solving Techniques

Problem-solving in algebra often involves formulating and solving equations based on a real-world scenario. In the ticket sales example, creating equations required translating the given information into algebraic expressions:

• The total number of tickets sold gave us one equation relating the number of adult and child tickets.
• The total amount of money collected provided another equation relating the cost of tickets and the number sold.

By identifying the variables (adult tickets, \(x\), and child tickets, \(y\)) and setting up equations based on these relationships, we turned a word problem into an algebraic one. Once the equations were formulated, we used the substitution method to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Navigating Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are all true at the same time. The aim is to find the value of each variable that satisfies all equations in the system.

In our theater problem, we worked with simultaneous linear equations representing ticket sales. The solution hinged on finding the values of \(x\) and \(y\) that would make both the total tickets and total revenue equations true at once. This interdependence is at the heart of simultaneous equations; changes in one variable affect the other.

Validating Solutions

After solving, it's crucial to check that the found values for \(x\) and \(y\) are sensible—e.g., they should be positive integers in the context of ticket sales. Additionally, substituting these values back into the original equations can verify their accuracy, ensuring that they reasonably represent the situation described in the problem.