Paulo's Investment Time Calculation Of Simple Interest

Hey guys! Let's dive into a math problem that's not just about numbers, but also about Paulo's smart financial move. We're going to break down a simple interest scenario step-by-step, so you can not only solve this particular problem but also understand the logic behind it. So, grab your thinking caps, and let's get started!

Unraveling the Simple Interest Concept

Simple interest, at its core, is a straightforward way of calculating interest on a principal amount. It's like the financial world's handshake – clear, direct, and easy to understand. Unlike compound interest, which adds the earned interest back into the principal, simple interest keeps things, well, simple. You earn interest only on the initial amount you invested or borrowed. This makes it a common choice for short-term loans and investments where the calculations need to be transparent.

The formula for simple interest is your trusty tool here: J = C * i * t. Let's break this down:

• J stands for the total interest earned. This is the reward you get for lending your money or the cost you pay for borrowing it.
• C is the principal amount, the initial sum of money you're dealing with. Think of it as the seed that grows into a larger amount.
• i represents the interest rate, expressed as a decimal. This is the percentage of the principal that you earn (or pay) as interest over a specific period.
• t is the time period for which the money is invested or borrowed, usually expressed in years or months. This is the duration of the financial handshake.

Understanding this formula is crucial. It's the key to unlocking not just this problem, but many others involving simple interest. Now, let's apply this knowledge to Paulo's investment.

Paulo's Financial Puzzle: Deciphering the Investment Details

Paulo, in his financial wisdom, invested R$25,000.00 in a simple interest operation. This is our principal amount (C), the seed money he planted. The interest rate was 3% per month. Now, remember, in our formula, we need to express the interest rate as a decimal. So, 3% becomes 0.03 (just divide 3 by 100). This is our interest rate (i), the monthly growth rate of Paulo's investment. At the end of the investment period, Paulo received R$2,250.00 in interest. This is our total interest earned (J), the fruits of his financial labor. The question we're trying to answer is: For how long was Paulo's money invested? This is the time period (t), the missing piece of our puzzle. We have three of the four variables in our simple interest formula (J = C * i * t). We know J (R$2,250.00), C (R$25,000.00), and i (0.03). Our mission is to find t. This is where the magic of algebra comes in. We'll rearrange the formula to solve for t, and then plug in the values we know. It's like being a financial detective, using clues to uncover the truth!

Cracking the Code: Solving for Time (t)

Alright, let's get our hands dirty with some math! We have the simple interest formula: J = C * i * t. Our goal is to isolate 't' on one side of the equation. This is a classic algebraic maneuver, like separating the yolk from the white of an egg. To do this, we'll divide both sides of the equation by (C * i). This gives us: t = J / (C * i). See how 't' is now all by itself? We've successfully rearranged the formula. Now comes the fun part: plugging in the values. We know J = R$2,250.00, C = R$25,000.00, and i = 0.03. Let's substitute these values into our formula: t = 2250 / (25000 * 0.03). First, we'll calculate the denominator: 25000 * 0.03 = 750. So, our equation becomes: t = 2250 / 750. Now, it's a simple division problem. 2250 divided by 750 equals 3. Therefore, t = 3. But hold on! Remember, the interest rate (i) was given as a monthly rate. This means our time period (t) is also in months. So, Paulo's money was invested for 3 months. Wait a minute! That doesn't seem to match any of the answer choices provided. Did we make a mistake? Let's double-check our calculations. Ah, it seems there was a small oversight in the initial problem setup. The interest earned (J) should have resulted in a different time period to align with the answer options. Let's proceed as if we were solving a similar problem with slightly adjusted numbers to illustrate the full process and then circle back to the correct answer based on the original data. Let's assume for a moment that the interest earned was R$6,750.00 instead of R$2,250.00. Plugging this new value into our formula, we get: t = 6750 / 750. Now, 6750 divided by 750 equals 9. So, if the interest earned was R$6,750.00, Paulo's money would have been invested for 9 months. This aligns with one of the answer choices (c), giving us a clear example of how to solve this type of problem. The key takeaway here is the process: rearrange the formula, plug in the values, and calculate the result. Even if the initial numbers need a slight tweak, the method remains the same. Now, let's refocus on the original problem with the correct interest earned of R$2,250.00 and determine the actual duration of Paulo's investment.

Refining the Calculation: The Correct Time Period

Okay, guys, let's get back to the original problem with Paulo's R$2,250.00 interest. We did the heavy lifting of setting up the formula and plugging in the values. We arrived at t = 2250 / 750. The correct calculation here is 2250 divided by 750, which equals 3. Oops! It seems we had a slight miscalculation earlier. So, t = 3 months. But hold on a second! Looking back at the original question, the options provided don't include 3 months. This indicates there might be a subtle misunderstanding of the question or a potential error in the provided answer choices. However, our calculation is solid based on the information given. Let's revisit the steps to ensure we haven't missed anything. We started with the simple interest formula: J = C * i * t. We identified the values: J = R$2,250.00, C = R$25,000.00, and i = 0.03 (3% per month). We rearranged the formula to solve for t: t = J / (C * i). We plugged in the values: t = 2250 / (25000 * 0.03). We calculated the denominator: 25000 * 0.03 = 750. We performed the division: t = 2250 / 750 = 3 months. Everything checks out! So, based on the provided information and our calculations, Paulo's money was invested for 3 months. It's possible that the answer choices are incorrect, or there might be additional context missing from the problem statement. In a real-world scenario, it's always a good idea to double-check the information and assumptions. For the sake of this exercise, we've accurately determined the time period based on the given data. Now, let's think about what we've learned and how we can apply it to other situations.

Mastering Simple Interest: Real-World Applications

So, guys, we've successfully navigated Paulo's investment puzzle. But the beauty of understanding simple interest is that it's not just about solving textbook problems. It's about equipping ourselves with a fundamental financial tool that's applicable in various real-world scenarios. Think about short-term loans, like those you might take out to cover a temporary cash flow gap. Simple interest is often used in these cases because it's easy to calculate and understand. Or consider investments like bonds, where you might earn a fixed rate of return over a specific period. Simple interest calculations can help you estimate your potential earnings. Even in everyday situations, understanding simple interest can be beneficial. For example, if you're lending money to a friend or family member, you might agree on a simple interest rate to ensure a fair return. The key takeaway is that simple interest is a building block for more complex financial concepts. By mastering the basics, you're setting yourself up for success in understanding and managing your finances. Now, let's recap the steps we took to solve Paulo's problem and highlight the essential principles. We started by understanding the concept of simple interest and its formula: J = C * i * t. This is our foundation, the bedrock upon which we built our solution. We then carefully identified the values given in the problem: the principal amount (C), the interest rate (i), and the total interest earned (J). This is like gathering the ingredients for a recipe. Next, we rearranged the formula to solve for the unknown variable, in this case, the time period (t). This is the cooking process, where we transform the ingredients into a delicious dish. We plugged in the known values and performed the calculation. This is the final step, where we get the answer. Finally, we interpreted the result in the context of the problem. This is the tasting part, where we ensure the dish is just right. By following these steps, you can confidently tackle any simple interest problem that comes your way. And remember, the more you practice, the more natural these calculations will become. So, keep flexing those financial muscles!

Conclusion: Paulo's Investment Time

In conclusion, by carefully applying the simple interest formula and working through the steps, we determined that Paulo's money was invested for 3 months. While this answer doesn't directly match the options provided, our calculations are based on the information given in the problem. This highlights the importance of both mathematical accuracy and critical thinking when dealing with real-world problems. We've not only solved a math problem but also reinforced our understanding of simple interest and its practical applications. So, the next time you encounter a financial scenario involving simple interest, you'll be well-equipped to tackle it with confidence! Keep practicing, keep learning, and keep those financial gears turning, guys!