Solving For R In The Formula A = P(1 + Rt)

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Given the formula $A=P(1+rt)$, where $A=27$ and $t=3$, solve for $r$.

This article provides a comprehensive guide on how to solve for the variable r in the formula A = P(1 + rt), given the values of A and t. This type of problem frequently appears in various mathematical contexts, particularly in financial calculations involving simple interest. Understanding the steps involved in isolating r is crucial for students and professionals alike. We will break down the solution process step-by-step, ensuring a clear understanding of each stage. Let's dive into the world of algebraic manipulation and master the art of solving for r.

Understanding the Formula

The formula A = P(1 + rt) is commonly used in simple interest calculations. Here's what each variable represents:

  • A represents the final amount or the accumulated value.
  • P represents the principal amount or the initial investment.
  • r represents the interest rate (expressed as a decimal).
  • t represents the time period (usually in years).

In this problem, we are given A and t, and our goal is to find r. The ability to manipulate formulas and solve for specific variables is a fundamental skill in algebra and has wide-ranging applications in finance, economics, and other fields. By understanding the relationships between these variables, we can solve for any unknown value, provided we have the others. The following sections will guide you through the process of isolating r and finding its value, given specific values for A, P, and t. This process not only enhances your algebraic skills but also deepens your understanding of the mathematical principles underlying simple interest and related calculations.

Problem Statement

We are given the formula:

A=P(1+rt)A = P(1 + rt)

We are also given that A=27A = 27 and t=3t = 3. Our objective is to solve for r. To do this, we will perform algebraic manipulations to isolate r on one side of the equation. This process involves several steps, each of which is crucial for arriving at the correct solution. First, we will distribute P across the terms inside the parentheses. Then, we will isolate the term containing r by subtracting P from both sides. Finally, we will divide by the coefficient of r to solve for r. By carefully following these steps, we can determine the value of r in terms of P. This process is a classic example of how algebraic principles can be applied to solve practical problems.

Step-by-Step Solution

Let's solve for r step-by-step:

  1. Start with the given formula: A=P(1+rt)A = P(1 + rt)

  2. Substitute the given values for A and t: 27=P(1+rimes3)27 = P(1 + r imes 3) 27=P(1+3r)27 = P(1 + 3r)

  3. Distribute P across the terms inside the parentheses: 27=P+3Pr27 = P + 3Pr This step is crucial as it separates the terms and allows us to isolate the term containing r. By multiplying P by both 1 and 3r, we transform the equation into a form where we can more easily manipulate it to solve for r. This distributive property is a fundamental concept in algebra and is essential for simplifying expressions and solving equations. Understanding how to apply the distributive property correctly is key to success in algebraic problem-solving.

  4. Isolate the term containing r by subtracting P from both sides: 27−P=3Pr27 - P = 3Pr Subtracting P from both sides ensures that the equation remains balanced while moving us closer to isolating r. This step is a direct application of the fundamental principle of algebraic manipulation: performing the same operation on both sides of an equation maintains its equality. By isolating the term containing r, we set the stage for the final step of solving for r. This step is a critical part of the process and demonstrates a clear understanding of how to manipulate equations to solve for unknown variables.

  5. Solve for r by dividing both sides by 3P: r = rac{27 - P}{3P} This final step isolates r and gives us the solution in terms of P. By dividing both sides of the equation by 3P, we effectively undo the multiplication and solve for r. This step is the culmination of the previous steps and provides the answer to the problem. The resulting expression for r shows its relationship to the given values of A and t, as well as the unknown value of P. This solution highlights the power of algebraic manipulation in solving for specific variables in complex equations.

Analyzing the Answer Choices

Now, let's compare our solution with the given answer choices:

  • A. r = rac{26}{3}
  • B. r = rac{27 - P}{3}
  • C. r = rac{27 - P}{3P}
  • D. r = rac{27 - 1}{3P}

Our derived solution is:

r = rac{27 - P}{3P}

Comparing this with the answer choices, we can see that Option C matches our solution.

Conclusion

Therefore, the correct answer is:

C. r = rac{27 - P}{3P}

Solving for a specific variable in a formula requires a clear understanding of algebraic principles and a systematic approach. In this case, we successfully isolated r by applying the distributive property, subtracting terms from both sides, and dividing by the coefficient of r. This process not only provides the correct answer but also reinforces essential algebraic skills that are applicable in various mathematical and real-world contexts. Mastering these techniques is crucial for anyone pursuing studies or careers in fields that require quantitative analysis and problem-solving.

This step-by-step solution demonstrates a clear and logical approach to solving algebraic equations. By understanding the underlying principles and applying them methodically, you can confidently tackle similar problems and enhance your mathematical proficiency.