Problem 58 - Elementary Algebra Within Reach

Question

In Exercises \(55-58\), write an algebraic equation. Do not solve the equation. An ice show earns a revenue of \(\$ 11,041\) one night. Tickets for the ice show cost \(\$ 18\) for adults and \(\$ 13\) for children. The number of adults attending the ice show was 33 more than the number of children attending the show. How many adults and children attended the show?

Step-by-Step Solution

Answer

The algebraic equation to represent the problem is \(18(x + 33) + 13x = 11041\).

Step 1: Define the Variables

Let's denote the number of children attending the show as \(x\). Since we know the number of adults attending the show was 33 more than the number of children, we can denote the number of adults as \(x + 33\).

Step 2: Write the Equation Based on Total Revenue

Since the total revenue from the show was $11,041 and adults tickets cost $18, while children tickets cost $13, we can write the total revenue as a sum of the revenues from the adults and children ticket sales. This gives us the equation \(18(x + 33) + 13x = 11041\).

Key Concepts

Defining Variables

Before solving any algebraic equation, it's crucial to clearly define what the variables represent. In this exercise, let's start by identifying the key information:

• The problem involves two groups: adults and children attending an ice show.
• We need to find out how many people attended, categorized by adults and children.
• The problem states a relationship between the number of adults and children: the adults are 33 more than the children.

To capture this relationship, we can define a variable for one of the groups. We'll choose to define the variable for children as the base. If we let the number of children be denoted by the variable \( x \), the adults can then be expressed as \( x + 33 \). This method of defining variables simplifies the problem as it creates a clear link between the unknowns and allows us to use one variable to describe both groups.

Writing Equations

Writing an equation involves translating the problem's conditions into a mathematical statement. In our problem, we're given the revenue generated and the ticket costs for both adults and children:

• Children's tickets cost \( \\(13 \) each.
• Adults' tickets cost \( \\)18 \) each.
• Total revenue for the night is \( \$11,041 \).

To write the equation:- Start by expressing the total revenue from children’s tickets: \( 13x \), where \( x \) is the number of children.- Next, express the total revenue from adults’ tickets: \( 18(x + 33) \).- Combine these to equal the total revenue: \[ 18(x + 33) + 13x = 11041 \].By setting up this equation, we represent the entire revenue situation mathematically, which is a critical step in algebraic problem solving.

Revenue Calculation

Understanding revenue calculation requires a good grasp of how individual components contribute to the whole. Here, revenue is generated from two ticket tiers:

• Children, each having a ticket price of \( \\(13 \).
• Adults, each having a ticket price of \( \\)18 \).

Revenue from children can be calculated by multiplying the number of children, \( x \), by the ticket price, \( \\(13 \), giving us \( 13x \).
Similarly, for adults, multiply the number of adults, \( x + 33 \), by \( \\)18 \), resulting in revenue from adults as \( 18(x + 33) \).
Combining these, the total revenue becomes the sum, so our equation for revenue, \( 18(x + 33) + 13x \), matches the observed revenue, equating to \( 11041 \). This breakdown helps in understanding the flow from individual components to total outcome, vital for accurate problem solving.

Problem Solving

Solving algebraic equations involves critical thinking and applying mathematical principles strategically. Although the problem instructs not to solve, dissecting how to approach solving it gives valuable insights.
First, you've defined variables linking adults and children. Then, using these, you've written a reliable equation.
When it comes to solving this kind of equation, it involves:

• Expanding the terms: distribute \( 18 \) into \( (x + 33) \).
• Combining like terms: gather all terms involving \( x \) on one side.
• Balancing the equation: keep the equation equal by performing operations on both sides.
• Isolating the variable \( x \) to find the number of children.
• Substitute back to find the number of adults.

Following these steps can lead to a solution. Practicing these processes consistently helps improve algebraic problem-solving skills and boosts confidence in tackling similar scenarios.