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Chapter 2: Problem 74

a. Solve \(A=P+P R T\) for \(R\). b. An investment of \(\$ 600\) grows to a future value of \(\$ 900\) in 5 years.Find the simple interest rate.

### Short Answer

Expert verified

The simple interest rate is 10\%.

## Step by step solution

01

## - Remove Parentheses and Simplify

The given equation is: \[A = P + PRT\] First, isolate the term involving \(R\): \[A - P = PRT\]

02

## - Solve for \(R\)

To isolate \(R\), divide both sides by \(PT\): \[R = \frac{A - P}{PT}\]

03

## - Substitute Values

Substitute \(A = 900\), \(P = 600\), and \(T = 5\) into the equation: \[R = \frac{900 - 600}{600 \times 5}\]

04

## - Perform Calculations

Calculate the numerator and denominator separately: \[R = \frac{300}{3000} = 0.1\]

05

## - Convert to Percentage

Convert the decimal form to percentage: \[R = 0.1 \times 100 = 10\%\]

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### Simple Interest Formula

To begin our discussion on simple interest, let's understand the basic formula. Simple interest is calculated using the formula:

\[A = P + PRT\]

Here,

- \(A\) represents the future value of the investment or loan, including interest.
- \(P\) is the principal amount, which is the initial amount of money invested or loaned.
- \(R\) stands for the annual interest rate (in decimal form).
- \(T\) corresponds to the time the money is invested or borrowed for, in years.

To solve for the interest rate \(R\), we need to rearrange the formula. This is essential, especially when you know the initial amount invested, the future value, and the duration of the investment.

###### Solving Equations

Now that we have the formula for simple interest, let's focus on solving equations to isolate \(R\). Start with the equation: \[A = P + PRT\]

First, we need to isolate the term that contains \(R\). Subtract \(P\) from both sides:

\[A - P = PRT\]

Next, to solve for \(R\), we will divide both sides of the equation by \(PT\):

\[R = \frac{A - P}{PT}\]

This isolates \(R\), and allows us to substitute the known values for \(A\), \(P\), and \(T\) to find the simple interest rate.

Let's apply this in our example where an investment of \(600 grows to \)900 in 5 years.

###### Percentages

Understanding percentages is crucial in finance and many other fields. The interest rate \(R\) found in the previous section will initially be in decimal form. Converting it into a percentage makes it easier to interpret.

After solving the equation, we found the decimal form of \(R\):

\[R = 0.1\]

To convert this decimal into a percentage, simply multiply by 100:

\[R = 0.1 \times 100 = 10\text{\textbackslash percent}\]

Hence, the simple interest rate is 10%. This percentage indicates how much interest the investment earns annually, relative to the original amount invested.

###### Investment Growth

Investment growth is a key motivation for understanding and applying the simple interest formula. It helps investors estimate how much their money will grow over time.

In our example, an initial investment of \(600 grows to \)900 over 5 years. By using the simple interest formula, we can find out how much interest is earned and what the annual interest rate is.

Here, the future value \(A\) is \(900, the principal \(P\) is \)600, and the time \(T\) is 5 years. Solving for \(R\) gives us a 10% annual simple interest rate.

Understanding simple interest and its calculations is fundamental for making informed financial decisions, whether you are investing money or borrowing it.

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