Problem 60 - Precalculus Mathematics for Calculus

Question

The admission fee at an amusement park is \(\$ 1.50\) for children and \(\$ 4.00\) for adults. On a certain day, 2200 people entered the park, and the admission fees that were collected totaled \(\$ 5050 .\) How many children and how many adults were admitted?

Step-by-Step Solution

Answer

1500 children and 700 adults were admitted.

Step 1: Define Variables

Let the number of children be represented by \( c \) and the number of adults be represented by \( a \). These are the two unknowns we need to solve for.

Step 2: Set Up Equations

We have two pieces of information that can be represented as equations:- The total number of people: \( c + a = 2200 \).- The total admission fees: \( 1.50c + 4.00a = 5050 \).

Step 3: Solve First Equation for One Variable

From the equation \( c + a = 2200 \), solve for \( c \):\[ c = 2200 - a \]

Step 4: Substitute into Second Equation

Substitute \( c = 2200 - a \) into the second equation:\[ 1.50(2200 - a) + 4.00a = 5050 \]

Step 5: Simplify and Solve for 'a'

Distribute and simplify the equation:\[ 3300 - 1.50a + 4.00a = 5050 \]Combine like terms:\[ 2.50a = 5050 - 3300 \]\[ 2.50a = 1750 \]Divide both sides by 2.50 to find \( a \):\[ a = \frac{1750}{2.50} = 700 \]

Step 6: Find 'c' Using 'a'

Substitute \( a = 700 \) back into \( c = 2200 - a \):\[ c = 2200 - 700 = 1500 \]

Step 7: Conclusion and Verification

The number of children is 1500 and the number of adults is 700. Verify by checking the total fees:\[ 1.50 \times 1500 + 4.00 \times 700 = 2250 + 2800 = 5050 \], which matches the given condition.

Key Concepts

Linear Equations

Linear equations are a fundamental component of algebra that allow us to represent and solve problems mathematically. A linear equation is an equation that forms a straight line when graphed on a coordinate plane and is defined by its constant terms and coefficients. The general form of a linear equation in two variables is:\[ ax + by = c \]where:

• \( x \) and \( y \) are variables, which represent unknown values we need to determine.
• \( a \) and \( b \) are coefficients, indicating how much each variable contributes to the equation's sum.
• \( c \) is a constant term, representing the total or target value in the context of the problem.

In the amusement park problem, we have two linear equations:

• One for the total number of people: \( c + a = 2200 \)
• Another for the total admission fee collected: \( 1.50c + 4.00a = 5050 \)

These equations allow us to use algebraic methods to find the number of children and adults entering the park.

Substitution Method

The substitution method is a useful technique for solving systems of linear equations. The idea behind substitution is to solve one equation for one variable, and then substitute that expression into another equation. This simplifies the system to a single equation in one variable and allows us to find the value of one unknown at a time.Here's how substitution was applied in the problem:

• First, solve one of the equations for one variable: We start with \( c + a = 2200 \) and solve for \( c \): \( c = 2200 - a \).
• Then, substitute \( c = 2200 - a \) into the other equation \( 1.50c + 4.00a = 5050 \).

By substituting, we eliminate the variable \( c \) and create an equation with just \( a \). This makes it easier to solve, as we can now focus on finding the value of \( a \) directly.

Problem-Solving

Problem-solving with systems of equations involves several steps that you can apply systematically. To find solutions effectively, follow these steps:1. **Understand the Problem:** Begin by reading the problem carefully to identify what is being asked and the data provided. We identified the task as finding the number of children and adults, given the total number of people and collected fees.
2. **Define Variables:** Introduce variables to represent the unknown quantities. In our case, \( c \) for children and \( a \) for adults.
3. **Formulate Equations:** Use the data provided in the problem to set up equations. Each piece of information acts as a building block. We had two key equations based on total number and total fees.
4. **Solve the System:** Apply algebraic methods such as substitution or elimination to find values for the variables. In this problem, substitution helped simplify and solve the equations.
5. **Verify the Solution:** Always check your work by plugging the values back into the original equations to ensure they satisfy all conditions.

The step-by-step process not only helps in obtaining the answer but also strengthens your understanding of algebra and logical thinking necessary in many other math and real-life scenarios.