Problem 17 - Prealgebra and Introductory Algebra

Question

Solve the formula for the given variable. $$A=P+P r ; P \quad \text { (Business) }$$

Step-by-Step Solution

Answer

The solution to the equation is \(P = \frac{A}{1+r}\)

Step 1: Set Up the Equation

We start by writing down the equation: \(A = P + Pr\)

Step 2: Group Terms with P

In this step, take \(P\) as a common factor for the terms on the right-hand side of the equation. The equation becomes \(A = P(1+r)\)

Step 3: Isolate P

The last step is to isolate the variable \(P\) by dividing both sides of the equation by \(1 + r\). The final equation becomes \(P = \frac{A}{1+r}\)

Key Concepts

Equation Solving

Equation solving is a crucial mathematical skill that allows us to find unknown values by manipulating equations. In the context of the exercise, we are given the equation \( A = P + Pr \) and asked to solve for \( P \).

To tackle problems like this, we follow a systematic approach:

• Identify the variable you need to solve for, ensuring clarity on what you are trying to find.
• Carefully rearrange and simplify terms to make the solving process smoother.
• Use operations such as addition, subtraction, multiplication, division, or even taking common factors to eliminate terms or isolate the desired variable.

In our specific example, we first rewrite the equation to make the term \( P \) easy to isolate. Understanding such processes is essential in both academic exercises and practical applications.

Isolate Variable

Isolating a variable involves rearranging an equation so that the variable stands alone on one side. This is a pivotal step in finding the solution to an equation.

In our exercise, the goal is to solve for \( P \) from the equation \( A = P + Pr \). We approach this by:

• Taking \( P \) as a common factor from the terms on the right-hand side, we have \( A = P(1 + r) \).
• Next, we need to get \( P \) alone. We do this by dividing both sides of the equation by \( 1 + r \), leading to \( P = \frac{A}{1+r} \).

This step is all about clarity and precision in making sure the equation balances while achieving our goal to isolate \( P \). Mastery of isolating variables is essential for solving many types of mathematical equations.

Business Mathematics

Business mathematics involves using mathematical methods to solve financial and business-related problems. The formula \( A = P(1 + r) \) appears frequently in business scenarios, particularly when dealing with interest and investment calculations.

Here’s how this formula is often applied in a business context:

• \( A \) represents the total amount after interest.
• \( P \) is the principal, or the initial amount of money or investment.
• \( r \) stands for the interest rate applied to the principal.

Understanding how to manipulate and solve such equations is vital for tasks like calculating loan repayments, determining investment growth, or predicting future profits. By mastering these concepts, students can enhance their ability to make informed decisions in the business world.