Problem 36 - Prealgebra

Question

Stacey is selling tickets to the school play. The tickets are \(\$ 6\) for adults and \(\$ 4\) for children. She sells twice as many adult tickets as children's tickets and brings in a total of \(\$ 112 .\) How many of each kind of ticket did she sell?

Step-by-Step Solution

Answer

Stacey sold 7 children's tickets and 14 adult tickets.

Step 1: Identify Variables

Let \( x \) be the number of children's tickets sold. Since Stacey sells twice as many adult tickets as children's tickets, the number of adult tickets sold is \( 2x \).

Step 2: Set Up the Equation

Using the ticket prices, the total revenue from children's tickets is \( 4x \) and from adult tickets is \( 6 \times 2x = 12x \). The total revenue Stacey brought in is \( 112 \). Therefore, we can set up the equation: \[ 4x + 12x = 112 \]

Step 3: Solve the Equation

Combine like terms in the equation to simplify it:\[ 16x = 112 \]Next, divide both sides by 16 to find \( x \):\[ x = \frac{112}{16} = 7 \]

Step 4: Determine Number of Tickets

Since \( x = 7 \), Stacey sold 7 children's tickets. Because the number of adult tickets is twice that of children's, Stacey sold \( 2 \times 7 = 14 \) adult tickets.

Key Concepts

Equations with Variables

In this problem, we use variables to represent unknown quantities, which is a common technique in math problems. Here, we choose a variable, \( x \), to represent the number of children's tickets sold. This choice is crucial because it helps us express other related quantities, such as the number of adult tickets sold, using the same variable.

By setting \( x \) as the children's tickets, we can write the number of adult tickets as \( 2x \), since twice as many adult tickets were sold. This makes it easier to manage and solve the problem since everything is in terms of one variable.

Variables allow us to translate word problems into mathematical equations. They act as placeholders that can take on different numerical values based on the scenario presented. This approach simplifies complex problems by focusing on fewer unknowns at a time.

Ticket Sales Problems

Ticket sales problems often involve comparing quantities and prices to find out how many items were sold or how much total money was made. In this exercise, we are given two types of tickets – adult and children’s – and their respective prices.

Stacey sells her tickets at \( \\(6 \) for adults and \( \\)4 \) for children. These values are crucial for setting up equations because they provide a way to calculate the total revenue. The key is to express the number of tickets sold not just in terms of themselves, but in terms of other variables.

Understanding the relationship between the items involved is important in solving ticket sales problems. Here, Stacey sells twice as many adult tickets as children's tickets, meaning the relationship between the numbers of tickets should be embedded in the equations you set up.

• Calculate the total money from children's tickets: \( 4x \).
• Calculate the total from adult tickets: \( 12x \).
• Combine these to understand the entire revenue: \( 4x + 12x = 112 \).

Solving Linear Equations

Linear equations are mathematical statements with an unknown that can be graphically represented as a straight line. Solving them involves finding the value of the unknown variable that makes the equation true.

In this problem, we combine the equation from the ticket revenues: \( 4x + 12x = 112 \). By merging like terms, \( 4x \) and \( 12x \), we get \( 16x = 112 \).

The next step is solving for \( x \) by isolating it. We divide both sides of the equation by 16 to simplify:

• Divide 112 by 16 to get \( x = 7 \).

This means Stacey sold 7 children's tickets. Knowing the solution for \( x \) allows us to find related values. Because she sold twice as many adult tickets, we calculate \( 2 \times 7 = 14 \) adult tickets.

Linear equations offer a straightforward way to connect different values and solve word problems involving relationships and totals. By practicing step-by-step solving, you become adept at breaking down even the most complicated problems.