Problem 67 - College Algebra Essentials

Question

Solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$s=P+P r t \text { for } r$$

Step-by-Step Solution

Answer

The rearranged formula to solve for the variable \( r \) is \( r = \frac{s - P}{Pt} \).

Step 1: Understand the formula

This is the simple interest formula. \( s \) is the total amount after time \( t \), \( P \) is the principal amount, \( r \) is the interest rate and \( t \) is the time the money is in for. We need to isolate \( r \).

Step 2: Subtract \( P \) from both sides of the equation.

To isolate \( r \), first subtract \( P \) from both sides of the equation to eliminate \( P \) from the right-hand side. We get \( s - P = Prt \).

Step 3: Divide both sides by \( Pt \)

Finally, we can isolate \( r \) by dividing both sides of the equation by \( Pt \). We get \( r = \frac{s - P}{Pt} \).

Key Concepts

Simple Interest Formula

The simple interest formula is a straightforward way to calculate the interest earned or paid on a specific principal amount over a period of time. This formula is particularly useful for short-term loans or investments where compounding isn't a significant factor.
In its basic form, the simple interest formula is expressed as:

• \( s = P + P r t \)

Here, \( s \) represents the total amount after interest, \( P \) is the principal (the initial amount of money), \( r \) is the interest rate, and \( t \) is the time period.
What makes this formula 'simple' is its linearity—interest is calculated only on the principal, leading to a constant rate of growth. This contrasts with compound interest, where interest itself is allowed to earn further interest.

Interest Rate Calculation

Calculating the interest rate from the simple interest formula can sometimes seem tricky, but with algebra, it's entirely manageable. Given the formula \( s = P + Prt \), you may need to find \( r \), the interest rate, which involves a few steps.
First, we need to recognize that \( s \) is already affected by the principal amount \( P \). Therefore, we subtract \( P \) from \( s \) to eliminate the initial principal and focus only on the interest earned or paid:

• \( s - P = Prt \)

Once we do this, the next step is to isolate \( r \). To do that, divide both sides of the equation by \( Pt \):

• \( r = \frac{s - P}{Pt} \)

Now, \( r \) is expressed in terms of the known quantities \( s \), \( P \), and \( t \). This simple manipulation allows us to calculate the interest rate effectively using basic algebra.

Algebraic Manipulation

Algebraic manipulation is all about reshaping equations to find unknown variables. It is one of the vital tools in mathematics, used to solve for variables in formulas.
In dealing with the simple interest formula, the task is to isolate the interest rate \( r \) by manipulating the original equation. We have these fundamental steps:

• Start with the equation: \( s = P + Prt \)
• Subtract \( P \) from both sides: \( s - P = Prt \)
• Divide by \( Pt \) to solve for \( r \): \( r = \frac{s - P}{Pt} \)

Every step involves an operation that maintains the equation's balance. Subtracting the same value from both sides or dividing both sides by the same value ensures that the equality holds, allowing us to confidently isolate and identify the variable in question. Thus, algebraic manipulation grants us a systematic approach to solve equations easily and accurately.