Problem 179 - Elementary Algebra

Question

The school play sold \(\$ 550\) in tickets one night. The number of \(\$ 8\) adult tickets was 10 less than twice the number of \(\$ 5\) child tickets. How many of each ticket were sold?

Step-by-Step Solution

Answer

30 child tickets and 50 adult tickets were sold.

Step 1 - Define the Variables

Let the number of child tickets be represented by \( x \). The number of adult tickets sold can be represented as \( 2x - 10 \) because the number of adult tickets is 10 less than twice the number of child tickets.

Step 2 - Set Up the Equation

The total amount of money from child tickets is \( 5x \) and from adult tickets is \( 8(2x - 10) \). We need to set up an equation that represents the total amount of money collected: \[ 5x + 8(2x - 10) = 550 \]

Step 3 - Simplify the Equation

Expand and simplify the equation: \[ 5x + 16x - 80 = 550 \] Combine like terms to get: \[ 21x - 80 = 550 \]

Step 4 - Solve for \( x \)

Add 80 to both sides of the equation to solve for \( x \): \[ 21x = 630 \] Divide both sides by 21: \[ x = 30 \]

Step 5 - Find the Number of Adult Tickets

Now that we know the number of child tickets \( x \) is 30, substitute \( x \) into the expression for adult tickets: \[ 2x - 10 = 2(30) - 10 = 60 - 10 = 50 \] So, 50 adult tickets were sold.

Step 6 - Verify the Solution

Verify the solution by calculating the total sales: \[ 30 \times 5 = 150 \] from child tickets and \[ 50 \times 8 = 400 \] from adult tickets. Adding these amounts gives: \[ 150 + 400 = 550 \], which matches the total sales.

Key Concepts

defining variables

In algebra word problems, it's necessary to define variables clearly so you know what you are working with. For example, in this exercise, we need to find out how many child and adult tickets were sold to make a total of $550.
To define the variables, follow these steps:

• Identify what you need to find.
• Assign variables to these unknowns.

Here, you would let the number of child tickets be represented by \( x \). Since the number of adult tickets is 10 less than twice the number of child tickets, the number of adult tickets is represented by \( 2x - 10 \).
Defining the variables like this makes it easier to set up and solve the equations.

setting up equations

Once you have defined your variables, the next step is to set up equations to represent the relationships described in the problem. In our example, we know the total sales and the price of each type of ticket.

• Calculate the total income from child tickets, which is \( 5x \)
• Calculate the total income from adult tickets, which is \( 8(2x - 10) \)
• Set up the equation that represents the total sales: \[ 5x + 8(2x - 10) = 550 \]

By organizing your information into an equation, you can then use algebraic techniques to find the values of the variables.

solving linear equations

With the equation set up, we now need to solve it. This involves simplifying the equation and isolating the variable. Here’s how:

• Expand and simplify the equation: \[ 5x + 16x - 80 = 550 \]
• Combine like terms: \[ 21x - 80 = 550 \]
• Add 80 to both sides: \[ 21x = 630 \]
• Divide both sides by 21: \[ x = 30 \]

Now, we know that 30 child tickets were sold.

verifying solutions

After solving for the unknowns, it's important to verify that your solution is correct. This ensures that your calculations match the problem's conditions.
To verify:

• Use the value of \( x \), which is 30, to find the number of adult tickets: \( 2x - 10 = 2(30) - 10 = 50 \)
• Calculate total sales: \( 30 \times 5 = 150 \) from child tickets and \( 50 \times 8 = 400 \) from adult tickets.
• Add these amounts: \( 150 + 400 = 550 \).

This matches the total given in the problem, confirming our solution of 30 child tickets and 50 adult tickets is correct.