Problem 49 - Elementary Algebra

Question

For exercises 49-52, the formula \(C=\frac{P_{m} P_{i}}{T F}\) describes the cost of insurance, \(C\). Is the relationship of the given variables a direct variation or an inverse variation? $$ P_{i}, T \text {, and } F \text { are constant; the relationship of } C \text { and } P_{m} \text {. } $$

Step-by-Step Solution

Answer

Direct variation

Step 1 - Understand the given formula

The formula given is \[ C = \frac{P_{m} P_{i}}{T F} \] where \( C \) represents the cost of insurance, \( P_{m} \) is some premium measure, \( P_{i} \) is a constant, \( T \) is a constant, and \( F \) is a constant.

Step 2 - Identify constants and variables

According to the problem, \( P_{i} \), \( T \), and \( F \) are constants. Thus, the only variables in consideration here are \( C \) and \( P_{m} \).

Step 3 - Simplify the equation

Since \( P_{i} \), \( T \), and \( F \) are constants, they can be combined into a single constant, let's call it \( k \). Rewrite the formula: \[ C = \frac{P_{m} \times k}{k} \] which simplifies to \[ C = k \times \frac{P_{m}}{k}. \] Further simplifying, we get \[ C = \frac{P_{m} P_{i}}{T F} \].

Step 4 - Analyze the equation

In the simplified equation, \( C = \frac{P_{m} \text{ * constant}}{\text{ * constant}} \), this shows that \( C \) directly varies with \( P_{m} \). If \( P_{m} \) increases, \( C \) also increases proportionally and vice versa.

Step 5 - Conclusion on variation type

Since \( C \) and \( P_{m} \) change proportionately and their relationship is positive, this indicates that the relationship is a direct variation.

Key Concepts

Direct Variation

Direct variation is a type of relationship between two variables where they change proportionally. This means if one variable increases, the other one also increases. Similarly, if one decreases, the other decreases as well. For instance, if mass increases in an object, the weight also goes up proportionally.

• In our exercise, the formula is given as \(C = \frac{P_{m} P_{i}}{T F}\).
• We are focusing on the relationship between \(C\), the cost of insurance, and \(P_{m}\), a premium measure.
• Here, \(P_{i}\), \(T\), and \(F\) are constants.

Since constants do not change, we can analyze how \(C\) adjusts with changes in \(P_{m}\). If \(P_{m}\) rises, \(C\) goes up in the same manner because they are multiplied directly. Therefore, the relationship between these variables is a direct variation.

Inverse Variation

Inverse variation describes a relationship where one variable increases as the other decreases. In simpler terms, if you multiply the two variables together, the result is always the same constant.

• For example, if speed goes up, the time taken to complete a journey goes down proportionally, provided the distance remains constant.
• In the formula \(C = \frac{P_{m} P_{i}}{T F}\), if we had an inverse variation scenario, a change in \(P_{m}\) would lead to an opposite change in \(C\).

However, in this exercise, we observe that \(C\) and \(P_{m}\) both move in the same direction, ruling out inverse variation.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They often include variables, constants, and arithmetic operations.

• In the provided exercise, \(C = \frac{P_{m} P_{i}}{T F}\) is an algebraic equation that relates the cost of insurance to various other factors.
• It is important to understand the role of each component in the equation.

In solving these types of equations, we isolate variables to understand how changes in one affect the other.

Constant Variables

In mathematical equations, constants are values that do not change. They provide a basis for comparison and calculation.

• In our exercise, \(P_{i}\), \(T\), and \(F\) are constants.
• They hold fixed values and do not alter with changes in other variables.

Understanding which factors are constants helps simplify the equation and focus on the variables that do change, in this case, \(C\) and \(P_{m}\). This simplification is crucial for determining the relationship between the variables.