Solving For R In The Formula A = P(1 + Rt)
In this comprehensive guide, we will delve into the process of solving for the variable r in the formula A = P(1 + rt), given the specific values of A = 27 and t = 3. This type of problem is a fundamental concept in algebra and often appears in various mathematical and financial applications. Understanding how to manipulate equations and isolate variables is a crucial skill for anyone studying mathematics, physics, engineering, or finance. Our main objective is to provide a clear, step-by-step explanation that ensures you not only arrive at the correct answer but also grasp the underlying principles. We'll break down each step, explain the reasoning behind it, and highlight common pitfalls to avoid. By the end of this article, you will confidently solve similar problems and have a solid foundation in algebraic manipulations. Whether you are a student looking to improve your algebra skills, a professional seeking a refresher, or simply someone curious about math, this guide is tailored to help you master this essential concept. This problem involves rearranging the equation to isolate r, which is a common task in algebra. We will walk through each step meticulously to ensure clarity and understanding. By solving this problem, you'll gain a deeper appreciation for the power and elegance of algebraic manipulation.
Problem Statement
We are given the formula:
where A = 27 and t = 3. Our goal is to solve for r.
This means we need to isolate r on one side of the equation. The process involves several algebraic steps, each of which we will detail below. Understanding these steps is key to solving not only this specific problem but also many other algebraic problems involving similar formulas. The ability to rearrange equations is a fundamental skill that is used extensively in various fields, including physics, engineering, economics, and computer science. By mastering this skill, you will be better equipped to tackle a wide range of problems that require mathematical analysis. The importance of understanding the underlying logic behind each step cannot be overstated; it's not just about getting the right answer but about developing a robust problem-solving approach. This article will guide you through the necessary steps, providing explanations and insights along the way.
Step-by-Step Solution
1. Substitute the Given Values
First, substitute the given values of A and t into the formula:
This substitution is the first concrete step in solving the problem. By replacing the variables A and t with their numerical values, we transform the equation into a more specific form that we can work with. This is a common technique in mathematics: substituting known values into an equation to simplify it and make it solvable. The resulting equation now contains only two variables, P and r, which makes it easier to isolate r. It’s important to perform this substitution accurately to avoid any errors in the subsequent steps. Double-checking the values and their placement in the equation is a good practice to ensure correctness. This step lays the foundation for the rest of the solution, so it’s crucial to get it right. By carefully substituting the values, we set ourselves up for a smooth and accurate solution process.
2. Distribute P
Next, distribute P across the terms inside the parentheses:
Distributing P involves multiplying it with both terms inside the parentheses: 1 and 3r. This step is based on the distributive property of multiplication over addition, a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac. Applying this property correctly is essential for simplifying expressions and solving equations. In this case, multiplying P by 1 gives P, and multiplying P by 3r gives 3Pr. This transformation is crucial because it helps us to separate the terms and move closer to isolating r. Distributing P makes it easier to manipulate the equation and rearrange the terms in the following steps. Accuracy in this step is paramount, as any mistake here will propagate through the rest of the solution. Therefore, it’s important to understand and apply the distributive property correctly.
3. Isolate the Term with r
Subtract P from both sides of the equation to isolate the term containing r:
This step is a fundamental algebraic manipulation technique used to isolate variables. By subtracting P from both sides of the equation, we maintain the equality while moving closer to isolating the term that contains r. This is based on the principle that if a = b, then a - c = b - c. The goal here is to get the term with r (which is 3Pr) alone on one side of the equation. This is a critical step in solving for r because it allows us to focus on the term that contains the variable we are trying to find. By isolating the term with r, we simplify the equation and prepare it for the final steps of the solution. It's important to perform the same operation on both sides of the equation to maintain balance and ensure the equality remains valid. This step is a key component of many algebraic problem-solving strategies.
4. Solve for r
Divide both sides by 3P to solve for r:
r = rac{27 - P}{3P}
This is the final step in isolating r and finding its value in terms of P. By dividing both sides of the equation by 3P, we effectively undo the multiplication by 3P that was affecting r. This is based on the principle that if a = b, then a/ c = b/ c, provided c is not zero. It’s crucial to ensure that P is not zero, as division by zero is undefined. This step directly isolates r on one side of the equation, giving us the solution we were looking for. The expression (27 - P)/(3P) represents the value of r in terms of P. This solution is general in the sense that it applies for any value of P (except zero). By performing this final division, we complete the process of solving for r and arrive at the answer. Understanding this step is essential for solving similar problems involving variable isolation.
Final Answer
Therefore, the solution for r is:
A. r = (27 - P) / (3P)
This final answer represents the value of r expressed in terms of P. It's the result of carefully applying algebraic principles to isolate r in the given equation. The solution demonstrates the importance of each step, from substituting the given values to distributing terms and isolating the variable. This process not only provides the correct answer but also reinforces the understanding of algebraic manipulation techniques. By arriving at this final answer, we confirm that the initial problem has been successfully solved, and the value of r has been determined. This solution is a testament to the power of algebra in solving real-world problems and mathematical challenges. It’s a concise and accurate representation of the relationship between r and P in the given context.
Conclusion
In conclusion, we have successfully solved for r in the equation A = P(1 + rt) given the values A = 27 and t = 3. The step-by-step process involved substituting the known values, distributing the terms, isolating the term containing r, and finally solving for r. The final solution, r = (27 - P) / (3P), provides a clear and concise answer to the problem. This exercise demonstrates the importance of algebraic manipulation skills in solving mathematical problems. The ability to rearrange equations and isolate variables is a fundamental skill that is applicable in various fields, including mathematics, physics, engineering, and finance. By mastering these techniques, you can tackle a wide range of problems that require mathematical analysis. This problem also highlights the significance of understanding the underlying principles behind each step. It’s not just about getting the right answer but about developing a robust problem-solving approach. By breaking down the problem into manageable steps and understanding the logic behind each step, we were able to arrive at the solution confidently. This approach can be applied to many other mathematical problems, making it a valuable skill for students and professionals alike.
