Problem 206 - Elementary Algebra

Question

Laurie was completing the treasurer's report for her son's Boy Scout troop at the end of the school year. She didn't remember how many boys had paid the \(\$ 15\) full-year registration fee and how many had paid the \(\$ 10\) partial-year fee. She knew that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. If \(\$ 250\) was collected for all the registrations, how many boys had paid the full-year fee and how many had paid the partial-year fee?

Step-by-Step Solution

Answer

4 boys paid the partial-year fee and 14 boys paid the full-year fee.

Step 1: Define the Variables

Let's define the variables for the number of boys who paid the full-year fee and the partial-year fee. Let \( x \) be the number of boys who paid the partial-year fee, and \( y \) be the number of boys who paid the full-year fee.

Step 2: Express the Relationship Between Variables

We know that the number of boys who paid for a full-year was ten more than the number who paid for a partial-year. This gives us the equation: \[ y = x + 10 \]

Step 3: Set Up the Total Fees Equation

We also know that the total amount collected from the registrations was \( \$ 250 \). The full-year fee was \( \$ 15 \) per boy and the partial-year fee was \( \$ 10 \) per boy. This gives us the equation: \[ 15y + 10x = 250 \]

Step 4: Substitute the First Equation into the Second

Substitute \( y = x + 10 \) from the first equation into the second equation: \[ 15(x + 10) + 10x = 250 \] This simplifies to: \[ 15x + 150 + 10x = 250 \]

Step 5: Solve for x

Combine like terms and solve for \( x \): \[ 25x + 150 = 250 \] Subtract 150 from both sides: \[ 25x = 100 \] Divide both sides by 25: \[ x = 4 \] So, 4 boys paid the partial-year fee.

Step 6: Solve for y

Since \( y = x + 10 \), we substitute \( x = 4 \) into the equation: \[ y = 4 + 10 \] \[ y = 14 \] So, 14 boys paid the full-year fee.

Key Concepts

Defining Variables

When tackling any algebra word problem, defining variables is the first step. Variables are letters we use to represent unknown quantities. In this problem, we want to find out how many boys paid the full-year versus the partial-year fee. So, we define:

• x: the number of boys who paid the partial-year fee
• y: the number of boys who paid the full-year fee

By defining these variables, we set a foundation for forming equations that describe the situation.

Setting Up Equations

After defining variables, we need to set up equations based on the information given.
First, we know the number of boys who paid the full-year fee is ten more than the number who paid the partial-year fee. This relationship can be written as:

• y = x + 10

Next, we use the financial data. If the total collected is \(\$ 250\), and each partial-year fee is \(\$ 10\), and each full-year fee is \(\$ 15\), we form the equation:

• 15y + 10x = 250

These equations help us encapsulate the problem in mathematical terms.

Solving Linear Equations

To find the values of the variables, we solve the linear equations we've set up.
Start with the two equations:

• y = x + 10
• 15y + 10x = 250

By substituting one equation into another or simplifying them, we'll isolate one variable and solve for it. The goal is to find concrete numerical values for x and y.

Substitution Method

The substitution method is a common technique in solving linear systems.
We use one equation to express a variable in terms of another variable, then substitute it into the second equation.
Given:

• y = x + 10
• 15(x + 10) + 10x = 250

We substitute \(y = x + 10\) into the second equation to get:

• 15(x + 10) + 10x = 250
• 15x + 150 + 10x = 250
• 25x + 150 = 250

By solving this, we find out that \(x = 4\).

Interpreting Solutions

Finally, interpreting the solutions means making sense of the values we found.
Remember our variables:

• x represents the number of boys paying the partial-year fee.
• y represents the number of boys paying the full-year fee.

From solving, we find:

• x = 4
• y = 14

Thus, 4 boys paid the partial-year fee, and 14 boys paid the full-year fee.Interpreting solutions connects the algebraic results back to the real-world context of the problem.