Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

The formula occurs in the indicated application. Solve for the specified variable. \(I=P r t\) for \(P\)

Step-by-Step Solution

Answer

The expression for \( P \) is \( P = \frac{I}{r t} \).

Step 1: Understanding the Formula

The formula given is the simple interest formula: \( I = P r t \), where \( I \) is the interest earned, \( P \) is the principal amount, \( r \) is the interest rate, and \( t \) is the time in years. We need to solve this equation for \( P \), the principal amount.

Step 2: Rearranging the Equation

To solve for \( P \), we need to isolate it on one side of the equation. Start with the given formula: \( I = P r t \). To isolate \( P \), divide both sides by \( r t \).

Step 3: Dividing Both Sides

By dividing both sides of the equation by \( r t \), we have: \[ P = \frac{I}{r t} \]. This rearrangement isolates \( P \), expressing it in terms of \( I \), \( r \), and \( t \).

Key Concepts

Solving Equations

When it comes to solving equations, you are applying rules and operations to find an unknown value. It starts by identifying what you need to solve — the variable representing the unknown. For this exercise, the unknown is the principal amount, denoted as \( P \). This process often entails rearranging equations. To help with this, we'll follow a systematic approach:

• Identify the equation and the variable to solve for.
• Use inverse operations to manipulate the equation, keeping operations balanced by performing the same step on both sides.
• Continue rearranging until the desired variable is isolated on one side of the equation.

In our case, the initial formula is \( I = Prt \). We want to isolate \( P \), so we divide both sides by \( rt \) to get \( P = \frac{I}{rt} \). Each step taken in solving an equation must adhere to the mathematical rule of performing equal operations on both sides, maintaining equilibrium throughout the process.

Financial Mathematics

Financial mathematics is a branch of applied mathematics concerned with financial problems. It covers a wide array of topics such as interest calculations, investments, loans, and more. Central to this field is the notion of the time value of money, which implies that a specific amount of money today has a different value in the future. This is where calculations involving interest become essential.In the simple interest formula \( I = Prt \), the variables interact to describe how interest grows linearly over time. Here:- \( P \) represents the principal, which is the initial amount of money.- \( r \) represents the interest rate, which must be expressed as a decimal for calculations.- \( t \) represents time, affected by the chosen duration for the investment or loan, typically tagged in years.Studying financial mathematics helps bridge the concepts of pure mathematics with practical financial applications, equipping you with the tools to make informed financial decisions.

Interest Calculations

Interest calculations help determine how much extra you will earn or pay over a period. Simple interest provides a straightforward approach to these calculations, where the interest is calculated only on the principal, not on accumulated interest.The simple interest formula is one of the fundamental components of financial literacy. By rearranging \( I = Prt \) to \( P = \frac{I}{rt} \), you can determine how much initial investment was required given the interest earned. This formula allows for:

• Ease in predicting and verifying the returns on fixed-income investments.
• Transparency in understanding loan repayments and associated costs.

Such calculations highlight the linear relationship between inputs and outcomes, making simple interest easy to grasp and apply. In more complex scenarios involving compound interest, the interest amount grows exponentially—much different from the constant growth seen in simple interest calculations.