Problem 13 - Prealgebra

Question

Complete each table. Determine whether the pattern forms a proportion. If so, write an equation relating the two quantities. Admission to an amusement park is 4 dollars plus 1.50 dollars per ride. $$\begin{array}{|l|l|l|l|l|}\hline \text { Number of rides } & 1 & & & \\\\\hline \text { Cost } & & & & \\ \hline\end{array}$$

Step-by-Step Solution

Answer

The pattern does not form a proportion since it includes a fixed cost. The table can be filled using the formula: Total Cost = 4 + 1.50 × Number of Rides.

Step 1: Understanding the Pattern

The cost pattern for the amusement park is composed of a fixed cost and a variable cost per ride. The fixed cost is the entry fee, which is $4, while the variable cost is $1.50 per ride.

Step 2: Calculate the Cost for Each Number of Rides

To find the total cost, use the formula \( \text{Total Cost} = 4 + 1.50 \times \text{Number of Rides} \). Compute the cost for the given number of rides. For 1 ride: \( 4 + 1.50 \times 1 = 5.50 \).

Step 3: Completing the Table

Using the formula, compute the cost for different quantities of rides: - For 1 ride: $5.50.- For 2 rides: \( 4 + 1.50 \times 2 = 7 \).- For 3 rides: \( 4 + 1.50 \times 3 = 8.50 \).- For 4 rides: \( 4 + 1.50 \times 4 = 10 \). Fill these values into the table.

Step 4: Determine If the Pattern Forms a Proportion

For a pattern to form a proportion, the ratio of the variables should be constant. Here, the cost does not increase linearly without a fixed starting point. As the pattern includes a fixed cost of $4, it does not form a proportion with the number of rides alone. Therefore, no equation of proportion exists.

Key Concepts

Proportion

In mathematics, a proportion is an equation that states that two ratios are equal. It implies a constant relationship between the quantities. For an amusement park scenario, if the costs increased proportionally with the number of rides, then after factoring in both the fixed and variable costs, it would form a proportion. In this case, the total cost consists of both a fixed entry fee and a per-ride fee. Since the fixed cost remains static regardless of the number of rides, the relationship between the number of rides and total cost is not purely proportional. Hence, no proportion exists here.

Variable Cost

Variable costs fluctuate with the level of output or activity. In the context of the amusement park, the variable cost is the cost per ride, which is $1.50. For each additional ride taken, the total cost increases by this amount.

• For 1 ride: Increase by $1.50
• For 2 rides: Further increase by $3.00
• And so on...

The variable cost is significant in determining the total expense, as it impacts how much more you spend with each ride taken.

Fixed Cost

A fixed cost is an expense that does not change with the level of goods or services provided. In this situation, the fixed cost is the $4 admission fee to the amusement park. This fee is paid once, regardless of the number of rides. It remains constant and forms the base cost that does not depend on how many rides you take. This ensures everyone starts with the same amount before factoring in additional ride costs.

Equation

An equation is a mathematical statement that asserts the equality of two expressions. For the amusement park scenario, the equation used to determine total cost is: \[ \text{Total Cost} = 4 + 1.50 \times \text{Number of Rides} \] This equation combines both the fixed and variable costs to calculate total expenses accurately. Utilizing this equation allows you to calculate how much you would pay based on the number of rides taken.

Pattern Recognition

Pattern recognition involves identifying regularities or trends in data. Recognizing patterns helps solve problems more quickly. In this exercise, by identifying a pattern in the amusement park costs, you can predict total expenses for any number of rides.

• Observe the constant $1.50 increase per ride.
• Notice the static $4 base cost regardless of rides.

Recognizing these patterns aids in forming equations and determining relationship types, such as understanding why this setup is not proportional but rather a fixed pattern plus variable increase.