Problem 61 - A Graphical Approach to College Algebra

Question

Solve each formula for the specified variable.} \(I=P R T\) for \(P \quad\) (Simple interest)

Step-by-Step Solution

Answer

\(P = \frac{I}{RT}\)

Step 1: Understand the Formula

The formula given is for simple interest: \(I = P R T\). This formula represents the interest \(I\) earned or paid over a period of time \(T\) on a principal amount \(P\) with an interest rate \(R\). The goal is to solve for the principal \(P\).

Step 2: Isolate the Variable

Since we need to solve for \(P\), we need to rearrange the formula to get \(P\) by itself. Start by dividing both sides of the equation by \(RT\):\[P = \frac{I}{RT}\].

Step 3: Write the Re-arranged Formula

After dividing both sides of the original equation by \(RT\), the re-arranged formula is:\(P = \frac{I}{RT}\). This formula indicates that the principal \(P\) is equal to the interest \(I\) divided by the product of the interest rate \(R\) and the time period \(T\).

Key Concepts

Understanding Simple Interest

Simple interest is a concept used in finance to calculate the additional amount earned or owed on money over time. This calculation is essential in many real-world scenarios, such as loans and investments.
At its core, the simple interest formula is straightforward:

• \[I = P \cdot R \cdot T\]

Here is what these symbols mean:

• \(I\) stands for interest, which is the money added to or owed in addition to the principal.
• \(P\) is the principal amount, or the initial sum of money.
• \(R\) represents the interest rate, which is usually given as a percentage.
• \(T\) is the time period over which the interest is calculated.

What makes simple interest 'simple' is that it is calculated only on the original principal, not on any interest that accumulates over time. Therefore, the calculation does not compound or increase over successive periods.

Solving Equations in Simple Interest

To solve equations, particularly those involving simple interest, one must be comfortable with manipulating algebraic expressions.
This involves performing operations like multiplication, division, addition, or subtraction to both sides of an equation to isolate unknown variables.
This is crucial when you know certain values of a formula and need to find an unknown value.In our original equation, we are solving for the principal \(P\).
The given formula is:

• \[I = P \cdot R \cdot T\]

To solve for \(P\), we rearrange the formula by isolating \(P\) on one side of the equation, which demands confidence in algebraic principles.

Isolating Variables in Algebra

Variable isolation is a fundamental skill in algebra.
It entails rearranging an equation to make a particular variable the subject, which is especially crucial when dealing with more complex problems.For the formula \(I = P \cdot R \cdot T\), to find \(P\), algebraic manipulation is necessary:

• First, identify the term to isolate: here, it's \(P\).
• Then, divide both sides by the product \(R \cdot T\) to get \(P\) alone:
  \[P = \frac{I}{R \cdot T}\]

This rearrangement tells us that the principal \(P\) is derived by dividing the interest \(I\) by the product of the rate \(R\) and time \(T\).Isolating variables like this allows one to express any variable in terms of others, a powerful tool for problem-solving.