Problem 21 - Algebra for College Students

Question

Solve each formula for the specified variable. $$\begin{aligned}&A=P(1+r t) \text { for } t\\\&\text { (future value for simple interest) }\end{aligned}$$

Step-by-Step Solution

Answer

\[t = \frac{\frac{A}{P} - 1}{r}\]

Step 1 - Identify the given formula

The formula provided is for calculating the future value of an investment using simple interest: \[A = P(1 + rt)\] where: - \(A\) is the future value - \(P\) is the principal amount - \(r\) is the rate of interest - \(t\) is the time period

Step 2 - Isolate the term with the variable \(t\)

To solve for \(t\), first isolate the term containing \(t\). Start by dividing both sides of the equation by \(P\): \[\frac{A}{P} = 1 + rt\]

Step 3 - Subtract 1 from both sides

Next, subtract 1 from both sides to get the term with \(t\) by itself: \[\frac{A}{P} - 1 = rt\]

Step 4 - Solve for \(t\)

Finally, to solve for \(t\), divide both sides of the equation by \(r\): \[t = \frac{\frac{A}{P} - 1}{r}\]

Key Concepts

Simple Interest Formula

The simple interest formula is used to calculate the future value of an investment or loan based on a fixed interest rate and time period. The formula is expressed as \[A = P(1 + rt)\]where:

• \(A\) is the future value of the investment or loan
• \(P\) represents the principal amount or initial investment
• \(r\) is the annual interest rate (expressed as a decimal)
• \(t\) is the time the money is invested or borrowed for, in years

This formula helps you understand how your money grows over time. It is fundamental in financial mathematics as it applies to various areas such as savings, loans, and investments. By mastering this formula, you can better predict financial outcomes and plan accordingly.

Isolate Variable

Isolating a variable means rearranging an equation to express a specific variable in terms of the others. In our given problem, we need to solve for the variable \(t\). We start with the formula: \[A = P(1 + rt)\]Here are the steps to isolate \(t\):

• First, divide both sides by \(P\) to isolate the term with \(t\): \[\frac{A}{P} = 1 + rt\]
• Then, subtract 1 from both sides: \[\frac{A}{P} - 1 = rt\]
• Finally, divide by \(r\) to solve for \(t\): \[t = \frac{\frac{A}{P} - 1}{r}\]

By isolating \(t\), we can determine the time it takes for the investment to reach a specified future value.

Algebraic Manipulation

Algebraic manipulation involves using arithmetic operations and rules to rearrange and solve equations. It is a fundamental skill in algebra and essential for solving for variables in formulas. Let's review the key steps in manipulating our simple interest formula:

• First, identify the formula and the variable you need to solve for.
• Use arithmetic operations like addition, subtraction, multiplication, and division to isolate the desired variable.
• Keep the equation balanced by performing the same operation on both sides.
• Carefully simplify each step, following the order of operations (PEMDAS/BODMAS).

Through these steps, we managed to isolate \(t\) in the simple interest formula: \[t = \frac{\frac{A}{P} - 1}{r}\]Algebraic manipulation allows us to derive meaningful insights from mathematical expressions and solve a wide range of problems.

Financial Mathematics

Financial mathematics applies mathematical methods to financial problems, enabling better decision-making and predictions. The simple interest formula falls under this category. It helps in understanding how investments grow over time. Here are some key applications:

• Savings Accounts: Calculate how much your money will grow in a savings account with a fixed interest rate.
• Loans: Determine the total amount repayable on a loan based on the principal, interest rate, and term.
• Investments: Plan future value of investments to achieve financial goals.

A good grasp of financial mathematics helps in making informed financial decisions, leading to better management of personal finances and investment strategies. By understanding and applying formulas like simple interest, you can efficiently plan for the future and achieve your financial objectives.