Solving For Simple Interest Find R In 816 = 600(1 + 9r)

Understanding simple interest is crucial for anyone looking to grow their savings or make informed financial decisions. This article breaks down the equation $816 = 600(1 + 9r)$, which represents the amount of money earned on a simple interest savings account. Our goal is to solve for $r$, the interest rate. We'll guide you through each step, ensuring you grasp the underlying concepts and can confidently tackle similar problems in the future.

Unpacking the Simple Interest Equation

The equation $816 = 600(1 + 9r)$ might seem daunting at first, but let's break it down piece by piece. This equation is a representation of the simple interest formula, which is a fundamental concept in finance. Understanding each component will make solving for $r$ much easier.

• 816: This represents the total amount of money you have at the end of the investment period. It includes the initial principal plus the earned interest. In financial terms, this is often referred to as the future value of the investment.
• 600: This is the principal, or the initial amount of money that was deposited into the savings account. It's the base upon which the interest is calculated. Think of it as the starting point of your investment journey.
• (1 + 9r): This is the core of the simple interest calculation. Let's dissect it further:
  • 1: Represents the original principal. It signifies that you're getting back the initial amount you invested.
  • 9: This is the time period, usually expressed in years. It indicates how long the money has been in the savings account. In our case, it's 9 years.
  • r: This is the simple interest rate, expressed as a decimal. It's the percentage of the principal that you earn as interest each year. Solving for $r$ is the main objective of this exercise.

By understanding these components, you can appreciate how the equation models the growth of money in a simple interest account. The equation essentially states that the final amount (816) is equal to the initial investment (600) plus the interest earned over 9 years at an interest rate of $r$. The simple interest formula is a linear equation, which makes it relatively straightforward to solve for any of its variables. Knowing the values of the other variables, we can isolate $r$ and determine the interest rate. This skill is invaluable for comparing different investment options and understanding the potential returns on your savings.

Before we dive into the algebraic steps, it's important to have a conceptual understanding of what we're trying to find. The interest rate, $r$, is the percentage return on our initial investment. A higher interest rate means more money earned over the same period. So, solving for $r$ will tell us how lucrative this particular savings account is. Furthermore, understanding how interest rates work is crucial for making informed decisions about savings, loans, and other financial products. The higher the interest rate, the more you earn on savings, but also the more you pay on loans. Therefore, mastering the concept of simple interest and how to calculate it is a fundamental skill in personal finance.

Step-by-Step Solution for 'r'

Now that we understand the equation and its components, let's walk through the steps to solve for $r$. We'll use basic algebraic principles to isolate $r$ on one side of the equation. Each step will be clearly explained to ensure you can follow along and replicate the process on your own.

1. Distribute the 600: Our first step is to distribute the 600 across the terms inside the parentheses. This means multiplying 600 by both 1 and 9r. This gives us: $816 = 600(1) + 600(9r)$ $816 = 600 + 5400r$ This step simplifies the equation by removing the parentheses and combining the constant terms. Distributive property is a cornerstone of algebra, and it's crucial for solving equations where terms are grouped together. In this case, it allows us to separate the constant term (600) from the term containing $r$ (5400r).

2. Isolate the term with 'r': Next, we want to isolate the term containing $r$ (5400r) on one side of the equation. To do this, we subtract 600 from both sides of the equation: $816 - 600 = 600 + 5400r - 600$ $216 = 5400r$ By subtracting 600 from both sides, we maintain the equality of the equation while moving the constant term to the left side. This step is a direct application of the subtraction property of equality, which states that you can subtract the same value from both sides of an equation without changing its solution. Isolating the term with the variable is a standard technique in solving algebraic equations.

3. Solve for 'r': Now, we have a simple equation with only one term containing $r$. To solve for $r$, we divide both sides of the equation by 5400: $216 / 5400 = 5400r / 5400$ $r = 216 / 5400$ This step utilizes the division property of equality, which is similar to the subtraction property. It allows us to isolate $r$ by dividing both sides of the equation by its coefficient (5400). The result, $r = 216 / 5400$, gives us the value of $r$ as a fraction.

4. Convert to decimal: The final step is to convert the fraction to a decimal. This will give us the value of $r$ in a more understandable format, as interest rates are typically expressed as decimals or percentages: $r = 216 / 5400 = 0.04$ To convert a fraction to a decimal, you simply divide the numerator (216) by the denominator (5400). The result, 0.04, is the decimal representation of the interest rate.

Therefore, we have solved for $r$ and found that the interest rate is 0.04. This means that the savings account earns 4% simple interest per year. This step completes the process of solving for r, and the resulting value provides valuable information about the investment. The interest rate is a key factor in determining the profitability of a savings account or other investment. A higher interest rate generally translates to a higher return on investment.

Verifying the Solution

It's always a good practice to verify your solution to ensure accuracy. We can do this by plugging the value of $r$ back into the original equation and checking if both sides are equal. This process helps to catch any potential errors made during the solving process and builds confidence in the result.

Let's substitute $r = 0.04$ into the original equation:

$816 = 600(1 + 9(0.04))$

Now, we simplify the right side of the equation:

$816 = 600(1 + 0.36)$ $816 = 600(1.36)$ $816 = 816$

Since both sides of the equation are equal, our solution $r = 0.04$ is correct. This verification step confirms that our calculated interest rate is consistent with the given information and the simple interest formula. Verification is an essential step in problem-solving, especially in mathematics and finance. It helps to ensure that the solution is accurate and that no errors were made during the process.

By verifying our solution, we have not only confirmed the accuracy of our calculations but also gained a deeper understanding of the relationship between the variables in the equation. We can see how the interest rate, time period, and principal all contribute to the final amount earned. This understanding is crucial for making informed financial decisions and for applying the simple interest concept to other real-world scenarios. In addition to verifying the solution, it's also helpful to consider the reasonableness of the answer in the context of the problem. For example, an interest rate of 4% per year is a realistic value for a simple interest savings account. If we had obtained a very large or very small value for $r$, it might indicate a potential error in our calculations.

Choosing the Correct Answer

Now that we've solved for $r$ and verified our solution, we can confidently choose the correct answer from the given options. We found that $r = 0.04$, which corresponds to option A.

A. $r = 0.04$ (Correct) B. $r = 0.14$ C. $r = 0.26$ D. $r = 0.40$

Option A is the correct answer because it matches the value of r that we calculated through the algebraic steps and verified by substituting back into the original equation. The other options represent different interest rates, but they do not satisfy the equation given in the problem. Choosing the correct answer is the final step in the problem-solving process, and it demonstrates a clear understanding of the concepts and procedures involved. In addition to choosing the correct answer, it's also important to understand why the other options are incorrect. This can help to reinforce the understanding of the underlying concepts and to avoid similar errors in the future. For example, options B, C, and D represent higher interest rates than the one we calculated. If we were to substitute these values into the original equation, we would find that the left side of the equation (816) would not equal the right side, indicating that these options are not correct.

Conclusion: Mastering Simple Interest Calculations

In this article, we've thoroughly explored how to solve for the interest rate ($r$) in the simple interest equation $816 = 600(1 + 9r)$. We broke down the equation, step-by-step, and demonstrated how to isolate $r$ using basic algebraic principles. We also emphasized the importance of verifying your solution to ensure accuracy and build confidence in your results.

By mastering simple interest calculations, you gain a valuable tool for understanding personal finance and making informed decisions about savings and investments. The ability to calculate interest rates, future values, and other financial metrics is essential for effective money management. Whether you're planning for retirement, saving for a down payment on a house, or simply trying to grow your savings, a solid understanding of simple interest will serve you well.

This skill extends beyond just solving equations. It empowers you to compare different investment options, understand loan terms, and make informed financial choices that align with your goals. Simple interest is a foundational concept in finance, and mastering it opens the door to more advanced topics such as compound interest, annuities, and investment strategies. By understanding the principles of simple interest, you can make better decisions about how to save, spend, and invest your money. This can lead to greater financial security and the achievement of your long-term financial goals.

Furthermore, the problem-solving skills we've used in this article are transferable to many other areas of life. The ability to break down complex problems into smaller steps, apply logical reasoning, and verify solutions is valuable in any field. The step-by-step approach we used to solve for r can be applied to a wide range of problems, from scientific calculations to everyday decision-making. By practicing these skills, you can enhance your critical thinking abilities and become a more effective problem solver in all aspects of your life. In addition to the practical benefits of understanding simple interest, there is also an element of empowerment that comes from taking control of your finances. By mastering financial concepts and developing a strong understanding of how money works, you can feel more confident in your ability to manage your finances and achieve your financial goals. This can lead to a greater sense of security and well-being, as well as the freedom to pursue your passions and live the life you want.

In conclusion, solving for $r$ in the simple interest equation is not just an exercise in algebra; it's a step towards financial literacy and empowerment. By understanding the concepts and practicing the skills presented in this article, you can take control of your financial future and make informed decisions that will benefit you for years to come.