Problem 55 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The accumulated amount after \(t\) yr when a deposit of \(P\) dollars is made in a bank and earning interest at the rate of \(r /\) year is \(A=P+P r t .\) Factor the expression on the right-hand side of this equation.

Step-by-Step Solution

Answer

The expression on the right-hand side of the equation A = P + Prt can be factored by identifying the common factor, P, and factoring it out, resulting in \(A = P(1 + rt)\).

Step 1: Identify the common factor

In the equation, A = P + Prt, observe that both terms on the right-hand side have P as a common factor. Step 2: Factor out the common factor

Step 2: Factor out the common factor

Factor out the common factor, P, from both terms on the right-hand side, giving: \(A = P(1 + rt)\). The expression on the right-hand side of the given equation has now been factored as \(A = P(1 + rt)\), where \(P\) is the common factor and \((1 + rt)\) represents the product of the interest rate and time plus 1.

Key Concepts

Accumulated Amount Formula

Understanding the accumulated amount formula is essential for those studying finance as it represents the total value, including principal and interest, after a certain period of time. When money is deposited into an account, it doesn't just sit there; it earns interest, and over time, the total amount grows.

The generic form of the accumulated amount formula is given by \( A = P + Prt \), where \( A \) represents the accumulated amount after time \( t \), \( P \) is the principal amount deposited, \( r \) is the annual interest rate (expressed as a decimal), and \( t \) is the time in years.

Let's break it down:

• \(\textbf{A} \): Accumulated Amount after \( t \) years.
• \(\textbf{P} \): Initial deposit (Principal).
• \(\textbf{r} \): Annual interest rate (as a decimal).
• \(\textbf{t} \): Time the money is invested or borrowed for in years.

When it comes to factoring, we look for common factors in an expression to simplify it. By identifying and extracting common factors, we can rewrite the expression in a more concise form, which in this case is \( A = P(1 + rt) \). This refactoring makes it easier to work with the equation and understand how each component contributes to the accumulated amount.

Simple Interest

Simple interest is one of the most straightforward concepts in finance, involving a quick calculation to determine how much interest is earned on a principal over time. Unlike compound interest, which calculates interest on the accumulated amount of previous periods, simple interest is only calculated based on the original principal.

The formula for simple interest is:\
\[ \text{Interest} = Prt \]
Using the elements we've already mentioned:

• \( P \): The principal amount (initial investment).
• \( r \): The annual interest rate.
• \( t \): The time the money is invested for in years.

If you deposit \(1,000 at a 5% annual interest rate for 2 years, the simple interest earned is:\
\[ Interest = 1000 \times 0.05 \times 2 = \)100 \]
With this in mind, it's evident that understanding and calculating simple interest is crucial, whether planning savings or understanding loans. The direct proportionality between time, principal, and interest rate in the context of simple interest makes it a critical element in financial literacy.

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra and are used to represent real-world problems in terms of mathematical symbols and operations. They consist of constants, variables, coefficients, and arithmetic operations (such as addition, subtraction, multiplication, and division).

In the context of our initial problem, \( A = P + Prt \) is an algebraic expression that involves variables \( A, P, r, \) and \( t \), with \( P \) being a common factor. Factoring algebraic expressions simplifies them, making them easier to manipulate and solve.

Here's why it's essential:

• It simplifies complex expressions, reducing the risk of errors in calculations.
• It provides a more straightforward form, which often reveals additional properties and relationships between the variables.
• It's a crucial step in solving equations, especially when isolating a variable or comparing two equations.

Understanding how to manipulate and factor algebraic expressions is a skill that underpins not just higher-level mathematics but numerous practical applications in science, engineering, economics, and beyond.