Problem 50 - 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? $$ P_{m}, T \text {, and } F \text { are constant; the relationship of } C \text { and } P_{i} $$

Step-by-Step Solution

Answer

Direct variation

Step 1: Identify the given formula

The formula given is: \[ C = \frac{P_{m} P_{i}}{T F} \]

Step 2: Understand the constants and variables

In the equation, \(P_{m}\), \(T\), and \(F\) are constants. We need to determine the relationship between \(C\) (cost of insurance) and \(P_{i}\) (one of the variables).

Step 3: Isolate the variable of interest

Rewriting the formula to focus on the relationship between \(C\) and \(P_{i}\), we get: \[ C \propto \frac{P_{i}}{T F} \]

Step 4: Analyze the proportion

Since \(P_{m}\), \(T\), and \(F\) are constants, the formula simplifies to: \[C \propto P_{i} \]

Step 5: Determine the type of variation

The result \(C \propto P_{i} \) depicts a direct variation, because \(C\) increases as \(P_{i}\) increases when all other variables are held constant.

Key Concepts

Insurance Cost Formula

The insurance cost formula provided in the exercise is:
\ \ \[ C = \frac{P_{m} P_{i}}{T F} \ \ \] This formula helps calculate the cost of insurance, denoted as \( C \). Here, \( P_{m}, T, \) and \( F \) are constants, and \( P_{i} \) is a variable. This formula highlights how various factors (premium amounts, time, and frequency) contribute to the overall insurance cost.
When analyzing such formulas, it's important to understand which terms are constants and which are variables. If we know that \( P_{m} \), \( T \), and \( F \) do not change, then we focus on \( P_{i} \) to see how it affects \( C \).
Constants like \( T \) (time) and \( F \) (frequency) can represent standardized periods and fixed rates, respectively. Hence, they help stabilize the equation, making \( P_{i} \) the primary influencer of \( C \).

Algebraic Equations

Algebraic equations allow us to express relationships between different quantities. In the context of the insurance cost formula, we see an algebraic equation that shows how different factors contribute to a specific outcome, the cost of insurance.
Analyzing the provided equation: \ \ \[ C = \frac{P_{m} P_{i}}{T F} \ \ \] we observe the relationship between the insurance cost \( C \) and another variable \( P_{i} \) while keeping \( P_{m}, T, \) and \( F \) constant.
By rearranging the equation, we isolate \( P_{i} \) to see its direct effect on \( C \). This process involves recognizing which elements are constant and how they simplify our understanding of the equation: \ \ \[ C \rightarrow \frac{P_{i}}{T F} \ \ \] This mathematical representation helps us conclude that \( C \) and \( P_{i} \) vary together. Understanding algebraic equations ensures we can solve for different variables and interpret the results in real-world contexts, such as insurance costs.

Direct Variation

The term 'direct variation' in algebra means that when one variable increases, the other variable increases proportionally, provided all other factors remain constant.
In the context of our insurance cost formula: \ \ \[ C = \frac{P_{m} P_{i}}{T F} \ \ \] isolating \( P_{i} \) shows a direct variation with \( C \): \ \ \[ C \rightarrow P_{i} \ \ \] This direct relationship implies that if \( P_{i} \) increases, \( C \) (insurance cost) also increases. Likewise, if \( P_{i} \) decreases, \( C \) will decrease.
Recognizing direct variation in equations helps when predicting how changes in one variable affect another. In practical scenarios, such as insurance, it's crucial to understand which factors influence the cost directly. This way, decisions around changing variables, such as altering premium amounts or coverage options, are well-informed and predictable.