Problem 21 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$S=P+P r t \text { for } r$$

Step-by-Step Solution

Answer

The formula rearranged to solve for 'r' (interest rate) is \(r = (S - P) / Pt\).

Step 1: Understanding the formula

Firstly, understand the formula \(S=P+Prt\) which represents the total amount owed after borrowing a principal (P) at an interest rate (r) over a time (t). In this scenario, the task is to solve this formula for the variable 'r'.

Step 2: Isolate 'r'

Isolate the 'r'. To do so, subtract 'P' from both sides of the equation to leave the 'Prt' term isolated on one side. The altered equation becomes \(S - P = Prt\).

Step 3: Solve for 'r'

To completely isolate 'r', divide both sides of the equation by 'Pt'. The final rearranged equation will be \(r = (S - P) / Pt\). This now gives 'r' as a function of 'S', 'P', and 't'.

Key Concepts

Algebraic Equations

Algebraic equations form the foundation of many mathematical concepts and are an essential part of solving real-world problems.

An algebraic equation is a statement of equality between two expressions that involve constants, variables, and arithmetic operations like addition, subtraction, multiplication, and division. The primary goal when working with these equations is to find the value of the unknown variables that make the equation true.

Recognizing and Solving Algebraic Equations

To solve an algebraic equation, the first step is to identify what you know and what you need to know—the knowns and the unknowns. For instance, in the equation given in our exercise, \(S = P + Prt\), 'S' represents the total amount after combining the principal and interest, whereas 'P', 'r', and 't' represent the principal, interest rate, and time, respectively.

In this exercise, the problem requires finding the value of the variable 'r', which is the interest rate. By manipulating the equation, we aim to express 'r' solely in terms of the other variables and known values.

Isolating Variables

Isolating a variable is a critical step in solving algebraic equations, as it allows you to find the value of that variable. The idea is to rearrange the equation so that the variable you're solving for is on one side of the equation by itself.

Steps to Isolate a Variable

To isolate a variable, perform inverse operations that 'undo' the way the equation is set up. This typically means using addition or subtraction to eliminate terms on one side and multiplication or division to clear away coefficients.

In the given exercise, isolating the variable 'r' involves two main steps: subtracting 'P' from both sides to obtain \(S - P = Prt\), and then dividing by 'Pt' to achieve \(r = (S - P) / Pt\), which effectively isolates 'r' on one side of the equation. Carefully executing these steps ensures that the integrity of the equation is maintained and the correct value of the isolated variable is found.

Interest Rate Formula

The interest rate formula is crucial for understanding how money accrues interest over time in financial contexts.

Components of the Interest Rate Formula

This formula typically includes the following components: the principal amount \(P\), the interest rate \(r\), and the time period \(t\) over which the interest is calculated. The formula expressed by \(S = P + Prt\) in our exercise is a simple representation of calculating the total amount \(S\) after interest is added to the principal.

When solved for 'r', as shown in the exercise, the formula \(r = (S - P) / Pt\) helps determine the interest rate based on the total amount accrued, the principal, and the time involved. Understanding and rearranging this formula correctly provides valuable insights in various financial scenarios, such as figuring out the interest rate on a loan or an investment.