Marina's Investment Journey Calculating Principal With Simple Interest
Hey guys! Today, we're diving into an exciting financial problem involving Marina, who has a clear goal in mind and a smart investment strategy. Let's break down the problem step by step, making sure we understand every detail so we can help Marina achieve her financial dreams. This is a classic example of how simple interest works, and by solving it, we'll gain valuable insights into managing our own investments.
Understanding the Problem
First off, let's clearly define Marina's financial goal. Marina aims to accumulate a total amount (montante) of R$ 187,900.00 within a specific timeframe of one year and nine months. She's already making her money work for her by investing her capital at a simple interest rate of 4.4% per month. The big question we need to answer is: How much capital did Marina initially invest to reach her goal? This involves understanding the mechanics of simple interest and applying the correct formulas to find the principal amount.
To get started, let's dissect the information we've been given. We know the future value (the montante), the time period, and the monthly interest rate. Our mission is to calculate the present value, or the principal, which is the initial amount Marina invested. We'll use the simple interest formula, which is a straightforward way to calculate interest earned on a principal amount over a certain period. Understanding each component of the formula will be key to unlocking the solution. We'll need to convert the time period into months to match the monthly interest rate, and then we'll be ready to plug the values into the formula and solve for the principal. This journey into financial mathematics will not only help Marina but also equip us with the tools to make informed decisions about our own financial futures. So, let's roll up our sleeves and get to work!
Decoding Simple Interest The Key to Marina's Success
Now, let's decode the magic of simple interest, guys! Simple interest is like the straightforward cousin of compound interest. It's calculated only on the principal amount, which makes it super easy to understand and predict. No complex calculations here – just a steady growth based on the initial investment. It's the financial equivalent of a slow and steady wins the race approach. For Marina, understanding simple interest is crucial because her investment is growing this way. The interest she earns each month is a fixed percentage of her original investment, and it doesn't compound, meaning she doesn't earn interest on previously earned interest. This simplifies the math but also means that the initial principal plays a huge role in the final amount she accumulates.
To really grasp this, let's break down the formula for simple interest. The formula is: M = P(1 + rt), where M is the montante (future value), P is the principal (initial investment), r is the interest rate per period, and t is the number of periods. Each of these components is interconnected, and understanding their relationship is key to solving our problem. The interest rate 'r' is usually expressed as a decimal, so we'll need to convert the percentage into a decimal form. The time period 't' needs to be in the same units as the interest rate period, which is why we'll convert the investment period into months. Once we have all these pieces aligned, we can plug them into the formula and solve for P, the principal amount. Understanding this formula isn't just about solving this problem; it's about building a foundation for understanding more complex financial concepts down the road. With a solid grasp of simple interest, we're ready to tackle the calculations and help Marina reach her financial goals.
Calculating the Investment Period Months Matter!
Before we jump into the numbers, let's nail down the investment period. Marina's timeline is one year and nine months, but to keep things consistent with the monthly interest rate, we need to convert this entire period into months. Think of it as translating the time into a language the interest rate understands. One year has 12 months, so one year and nine months is simply 12 months plus 9 months, which equals 21 months. It's a straightforward conversion, but it's a critical step. If we overlooked this, our calculations would be way off, and we'd end up with the wrong answer. This highlights the importance of attention to detail in financial calculations. Each number and unit plays a crucial role, and making sure they all align is paramount.
Why is this conversion so important? Well, the interest rate is given as a monthly rate (4.4% per month). To accurately calculate the total interest earned over the investment period, we need to express the period in the same units. Imagine trying to add apples and oranges – it doesn't work until you have a common unit, like fruit. Similarly, we need to express both the interest rate and the time period in monthly terms to perform the calculation correctly. This step also reinforces the idea that financial math isn't just about plugging numbers into formulas; it's about understanding the underlying concepts and ensuring that we're working with consistent units. By converting the investment period to months, we're setting ourselves up for accurate calculations and a successful solution to Marina's investment puzzle. So, with this key piece in place, we're ready to move on to the next step and start crunching the numbers!
The Grand Finale Unveiling Marina's Initial Investment
Alright, guys, it's calculation time! We've laid the groundwork, understood the concepts, and converted our units. Now, we're ready to put the pieces together and solve for the principal amount Marina invested. We'll be using the simple interest formula we discussed earlier: M = P(1 + rt). Remember, M is the montante (R$ 187,900.00), r is the monthly interest rate (4.4% or 0.044 as a decimal), and t is the time in months (21 months). Our mission is to find P, the principal.
Let's rearrange the formula to solve for P. We get: P = M / (1 + rt). Now, it's just a matter of plugging in the values. So, P = 187,900 / (1 + 0.044 * 21). First, we calculate the value inside the parentheses: 0.044 * 21 = 0.924. Then, we add 1: 1 + 0.924 = 1.924. Now we have: P = 187,900 / 1.924. Performing the division, we get P ≈ R$ 97,661.12. So, Marina initially invested approximately R$ 97,661.12 to reach her goal of R$ 187,900.00 in one year and nine months at a simple interest rate of 4.4% per month. This result aligns with option (C) in the problem statement.
This calculation demonstrates the power of understanding simple interest and applying the formula correctly. It also underscores the importance of breaking down a problem into smaller, manageable steps. We started by defining the goal, then deciphered the concept of simple interest, converted the time period, and finally, plugged the values into the formula to arrive at the solution. Each step was crucial, and together, they led us to the answer. With this understanding, we can confidently tackle similar financial problems and make informed decisions about our own investments. Marina's journey serves as a great example of how financial planning and a solid grasp of mathematical concepts can lead to achieving our financial goals.
Choosing the Correct Answer Why Option (C) Reigns Supreme
So, after our deep dive into the problem, we've arrived at the correct answer, and it's time to celebrate the victory! We meticulously worked through the calculations, and our final result for Marina's initial investment was approximately R$ 97,661.12. Now, let's connect this result to the answer choices provided in the question. We had four options:
- (A) R$ 20.335,49
- (B) R$ 42.704,54
- (C) R$ 97.661,12
- (D) R$
Comparing our calculated value (R$ 97,661.12) with the options, it's crystal clear that option (C) matches our result perfectly. Therefore, option (C) is the undisputed champion and the correct answer to the question. It's always a good practice to double-check our calculations and make sure that the answer we've derived aligns with one of the options provided. This serves as a sanity check and reinforces our confidence in the solution.
But, beyond just finding the correct answer, it's important to understand why the other options are incorrect. Options (A) and (B) are significantly lower than our calculated value, indicating a much smaller initial investment. If Marina had invested those amounts, she wouldn't have reached her goal of R$ 187,900.00 in the given timeframe with the given interest rate. This highlights the importance of a substantial initial investment when aiming for a significant financial target. Option (D) is not even a valid response, further solidifying (C) as the single choice, so by systematically working through the problem and carefully evaluating the answer choices, we can confidently select the correct answer and gain a deeper understanding of the underlying financial principles. This approach not only helps us solve this specific problem but also equips us with the skills to tackle future financial challenges with confidence and precision.
Wrapping Up Marina's Success Story and Our Learning Journey
Alright, guys, we've reached the end of our financial quest, and what a journey it has been! We started with a seemingly complex problem involving Marina's investment goal, but we broke it down into manageable steps, conquered each one, and emerged victorious. We successfully calculated the initial capital Marina invested, which was approximately R$ 97,661.12, making option (C) the correct answer. But more than just solving a problem, we've learned some valuable lessons about financial planning and the power of simple interest.
We've seen how understanding the fundamental concepts of simple interest can help us make informed decisions about our own investments. We've learned the importance of converting units to ensure consistency in our calculations. We've practiced rearranging formulas to solve for different variables. And, perhaps most importantly, we've reinforced the idea that breaking down a problem into smaller steps makes it much less daunting. Marina's story is a testament to the fact that with a clear goal, a solid understanding of financial principles, and a bit of math, we can all achieve our financial dreams.
So, what's the takeaway from all this? It's simple: financial literacy is a superpower! The more we understand about how money works, the better equipped we are to manage our finances and achieve our goals. This problem wasn't just about numbers; it was about empowering ourselves with knowledge. And with each problem we solve, each concept we grasp, we're building a stronger foundation for a brighter financial future. So, let's keep learning, keep practicing, and keep striving for financial success. Who knows? Maybe one day, we'll be sharing our own success stories, just like Marina's!
Simple interest, principal, investment, financial goal, calculation, monthly interest rate, time period, formula, montante, initial capital
