Problem 91 - College Algebra and Calculus: An Applied Approach

Question

Solve for the indicated variable. Investment at Simple Interest Solve for \(r\) in \(A=P+P r t\).

Step-by-Step Solution

Answer

The rate \(r\) in the formula for simple interest can be solved as \(r = \frac{A - P}{Pt}\).

Step 1: Rearranging the formula

Firstly, subtract \(P\) from both sides of the equation to isolate the terms that include \(r\). This yields \(A - P = Prt\).

Step 2: Isolate the variable r

Next, divide both sides of the equation by \(Pt\), which isolate \(r\). The final formula for \(r\) will be \(r = \frac{A - P}{Pt}\).

Key Concepts

Simple Interest Formula

The simple interest formula is a key concept in understanding how investments grow over time when the interest is not compounded. It calculates the interest earned or paid on a principal amount for a certain period and at a constant interest rate.

In essence, the formula for simple interest is written as:
\( I = Prt \)
Where:

• \( I \) represents the interest earned,
• \( P \) is the principal amount (the initial sum of money),
• \( r \) stands for the annual interest rate (in decimal form), and
• \( t \) is the time the money is invested for, in years.

The amount \( A \) at the end of the investment period would then be the sum of the principal plus the interest earned, expressed by the formula: \( A = P + I \) or \( A = P + Prt \), which simply combines the principal with the interest formula. Students can use this to determine how much money will be accumulated after investing at a simple interest rate for a set period.

Isolate a Variable

Learning to isolate a variable is a fundamental skill in algebra that allows us to find the value of one variable in terms of others. The process involves using various algebraic operations to get the variable of interest by itself on one side of the equation.

For instance, when solving an equation like \( A = P + Prt \) for the variable \( r \) which represents the interest rate, the goal is to perform operations that will leave \( r \) by itself on one side. To achieve this, follow these steps:

• Move all other variables and constants to the opposite side of the equation through addition or subtraction.
• If the variable is multiplied by other variables or divided by any number, counteract this by performing the opposite operation. This means multiply if it's divided or divide if it's multiplied.

By doing so, you effectively 'isolate' the variable you are trying to solve for, allowing you to calculate its value given the values for all other variables.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions and equations to solve for particular variables. This process is essential when solving for variables as it helps to transform complex problems into simpler, more solvable equations. A firm grasp of such manipulations allows you to find solutions that would otherwise be obscured by the original form of the equation.

Continuing with the example of solving for the variable \( r \) in the simple interest formula, algebraic manipulation consists of two main steps:

Rearranging the formula

First, terms are moved across the equal sign by carrying out inverse operations - subtracting \( P \) from both sides to have all terms containing \( r \) on one side, resulting in \( A - P = Prt \).

Isolating the variable

Next, dividing both sides by \( Pt \) counteracts the multiplication by \( Pt \) and leaves \( r \) on its own: \( r = \frac{A - P}{Pt} \).
Properly isolating the variable allows you to clearly see the relationship between the variable and other elements in the equation, providing you with a direct way to calculate its value when you have the necessary information.