Investment Return: Calculate Original Amount

Hey guys! Today, we're diving into a financial puzzle where we need to figure out the original investment amount based on the interest earned. Let's break down this problem step by step so we can all understand it clearly. This kind of problem is super relevant in real life, especially when you're planning your own investments. So, buckle up, and let's get started!

Understanding the Problem

Okay, so here’s the scenario: Andrés received $1324 in interest from an investment. This investment was made 6 years and 9 months ago, and the interest rate was 9.5% semiannually. Our mission, should we choose to accept it (and we do!), is to find out the original amount Andrés invested. This means we need to use the information about the interest earned, the time period, and the interest rate to work backward and calculate the principal amount. Think of it like being a financial detective – we have clues, and now we need to piece them together to solve the mystery!

To really nail this, we need to get cozy with a few key concepts. First off, we're talking about compound interest. This isn't your run-of-the-mill simple interest; compound interest means that the interest earned in each period is added to the principal, and the next interest calculation is based on this new, higher amount. It's like a snowball rolling down a hill – it gets bigger and bigger as it goes. This makes a huge difference over longer periods, so understanding it is crucial. Secondly, we need to be crystal clear on interest rates and periods. We're told the rate is 9.5% semiannually, which means it's applied every six months, not every year. This is a sneaky detail that can throw you off if you're not careful. We've got to convert everything into consistent units – years and annual rates – to avoid any calculation mishaps. And finally, we need to remember our financial formulas. There are formulas that directly relate principal, interest rate, time, and the final amount. Knowing which formula to use and how to rearrange it is the key to cracking this problem. So, before we jump into the calculations, let's make sure we're all on the same page with these foundational concepts. Got it? Let's move on!

Breaking Down the Given Information

Alright, let's get down to brass tacks and dissect the info we've got. This is like gathering all the pieces of a puzzle before we start putting it together. We need to be super clear on what we know and what we're trying to find. So, let's break it down:

• Interest Earned (I): $1324 – This is the extra cash Andrés pocketed from his investment. It's the reward for his patience and savvy financial planning!
• Time Period (t): 6 years and 9 months – Now, this is where we need to be a bit careful. We can't just waltz into our calculations with this mixed unit of years and months. We need to convert it all into one consistent unit, and years are usually the go-to for these kinds of problems. So, 9 months is like 9/12 of a year, which simplifies to 0.75 years. Add that to the 6 years, and we get a total time period of 6.75 years. See how we tidied that up? This attention to detail is crucial for getting the right answer.
• Interest Rate (r): 9.5% semiannually – Ah, the plot thickens! This isn't a straight-up annual rate, folks. Semiannually means it's applied every six months. So, to get our annual rate, we need to double it. That's 9.5% * 2, which gives us a sweet 19% annual interest rate. Boom! We've just converted a potentially confusing semiannual rate into a clear, usable annual rate. Remember, always double-check these details – they can make or break your calculations!

So, let's recap. We've got $1324 in interest, a time period of 6.75 years, and an annual interest rate of 19%. These are our key ingredients. Now, what's the unknown? We're hunting for the original investment amount, often called the principal (P). This is the amount Andrés initially plunked down, the seed that grew into the $1324 interest. Knowing our unknowns and knowns is half the battle. Now, we're armed and ready to choose the right formula and solve for P. Let’s keep going!

Choosing the Right Formula

Okay, team, now comes the crucial part where we pick the right tool for the job. In our case, the tool is a formula – the one that connects interest earned, principal, interest rate, and time. There are a few financial formulas floating around, but we need the one that fits our scenario perfectly. Remember, we're dealing with compound interest here, where interest gets added to the principal, and the next interest calculation is based on the new, higher amount. This is different from simple interest, where interest is calculated only on the original principal.

The formula we need is the compound interest formula, which looks like this:

A = P (1 + r/n)^(nt)

Where:

• A is the future value of the investment/loan, including interest.
• P is the principal investment amount (the initial deposit or loan amount).
• r is the annual interest rate (as a decimal).
• n is the number of times that interest is compounded per year.
• t is the number of years the money is invested or borrowed for.

But wait, there's a slight twist! We don't know A (the future value), but we do know the interest earned (I). So, we need to tweak our formula a bit to make it work for us. We know that the interest earned is the difference between the future value (A) and the principal (P). In other words:

I = A - P

Now, we can substitute A in the first formula with (I + P). This gives us:

I + P = P (1 + r/n)^(nt)

This is the magic formula we're going to use! It directly relates the interest earned (I), the principal (P), the interest rate (r), the number of compounding periods per year (n), and the time (t). See how we adapted the standard formula to fit our specific needs? This is a key skill in problem-solving – knowing how to adjust your tools to the situation at hand. Now that we've got our formula locked and loaded, let's plug in those numbers and solve for P. We're getting closer to cracking this case!

Plugging in the Values

Alright, detectives, it's time to put our formula to work! We've got all the pieces of the puzzle; now we just need to slot them into the right places. This is where careful attention to detail pays off – one wrong number, and our whole calculation could go haywire. So, let's take it slow and steady, making sure everything lines up perfectly.

Remember our tweaked compound interest formula? It's:

I + P = P (1 + r/n)^(nt)

Let's remind ourselves what each of these letters stands for and what values we know:

• I (Interest Earned): $1324
• P (Principal): This is what we're trying to find!
• r (Annual Interest Rate): 19%, which we need to convert to a decimal by dividing by 100, so r = 0.19
• n (Number of Compounding Periods per Year): Since the interest is compounded semiannually, that means it's compounded twice a year, so n = 2
• t (Time in Years): 6.75 years

Now, let's carefully plug these values into our formula. It'll look something like this:

$1324 + P = P (1 + 0.19/2)^(2 * 6.75)

See how we've replaced each letter with its corresponding value? This is a crucial step – it transforms our abstract formula into a concrete equation that we can actually solve. It might look a bit intimidating with all those numbers and parentheses, but don't worry, we'll tackle it step by step. We're not afraid of a little math, are we? Now that we've got everything plugged in, the next step is to simplify and solve for P. We're on the home stretch now – let’s keep that momentum going!

Solving for the Principal (P)

Okay, mathletes, the moment of truth has arrived! We've got our equation all set up, and now it's time to unleash our algebraic prowess and solve for P, the elusive principal amount. This part might seem a bit like a mathematical obstacle course, but don't sweat it – we'll break it down into manageable steps. Remember, the key is to take it one operation at a time and keep our eye on the prize: isolating P on one side of the equation.

Here's our equation again, just to refresh our memories:

$1324 + P = P (1 + 0.19/2)^(2 * 6.75)

First, let's simplify the stuff inside the parentheses and the exponent:

• 0. 19 / 2 = 0.095
• 1 + 0.095 = 1.095
• 2 * 6.75 = 13.5

So, our equation now looks like this:

$1324 + P = P (1.095)^13.5

Next, we need to calculate (1.095)^13.5. This is where a calculator comes in super handy! If you punch that in, you should get approximately 3.575. So, our equation becomes:

$1324 + P = 3.575P

Now, we need to get all the P terms on one side of the equation. Let's subtract P from both sides:

$1324 = 3.575P - P

This simplifies to:

$1324 = 2.575P

Finally, to isolate P, we'll divide both sides by 2.575:

P = $1324 / 2.575

Punch that into your calculator, and you should get approximately:

P = $514.17

Woohoo! We did it! We've successfully solved for P. This means that the original investment amount, the principal that Andrés invested 6 years and 9 months ago, was approximately $514.17. Give yourselves a pat on the back – you've just navigated a pretty complex financial calculation. But we're not done yet – the final step is to make sure our answer makes sense in the real world.

Verifying the Solution

Awesome job making it this far, everyone! We've crunched the numbers and arrived at an answer, but in the real world, it's crucial to double-check our work and make sure our solution makes sense. This isn't just about getting the right number; it's about understanding the context and ensuring our answer is logical within that context. So, let's put on our critical-thinking caps and give our solution a good once-over.

We calculated that Andrés's original investment was approximately $514.17. Now, let's ask ourselves: does this sound reasonable? He earned $1324 in interest over 6 years and 9 months with a 19% annual interest rate (compounded semiannually). That's a pretty substantial return, but given the relatively high interest rate and the length of the investment, it's not outlandish. If we had gotten an answer like $5 or $5000, we'd know something went seriously wrong along the way!

Another way to verify is to do a quick, rough calculation to see if our answer ballpark checks out. If Andrés invested around $500 at a 19% annual rate, he'd earn roughly $95 in interest in the first year. Because the interest is compounded, the interest earned each year would increase. Over 6.75 years, this could easily add up to $1324. This isn't a precise calculation, but it gives us a good sanity check.

To be super thorough, we could even plug our calculated principal back into the original compound interest formula and see if it spits out an interest earned close to $1324. This is like the ultimate test – if it passes, we can be pretty darn confident in our answer.

But here's the bigger picture: verifying the solution isn't just about math; it's a life skill. In any problem-solving scenario, whether it's financial, scientific, or even everyday decisions, it's essential to take a step back and ask, "Does this make sense?" This simple question can save you from countless errors and help you make sound choices. So, let’s give ourselves a big pat on the back for not only solving the problem but also for verifying our solution like true financial pros!

Conclusion

Alright, financial wizards, we've reached the finish line! We successfully navigated a complex problem involving compound interest, and we've emerged victorious. We started with a scenario where Andrés earned $1324 in interest from an investment made 6 years and 9 months ago at a 9.5% semiannual interest rate. Our mission was to uncover the original investment amount, and through careful analysis, strategic formula selection, and some good ol' fashioned math, we cracked the case!

We found that Andrés's original investment was approximately $514.17. This means that by investing this amount wisely and letting the magic of compound interest do its thing, Andrés was able to generate a significant return over time. This is a fantastic illustration of the power of long-term investing and the importance of understanding how interest works. You are awesome!

But more than just getting the right answer, we learned some valuable problem-solving skills along the way. We practiced breaking down a complex problem into smaller, more manageable parts. We identified the key information, chose the appropriate formula, and carefully plugged in the values. We even tweaked the standard formula to fit our specific needs – a crucial skill in any problem-solving scenario. And, most importantly, we emphasized the importance of verifying our solution to ensure it makes sense in the real world.

These skills aren't just for math class; they're for life! Whether you're planning your own investments, making financial decisions, or tackling any kind of challenge, the ability to analyze, strategize, and verify is essential. So, give yourselves a huge round of applause for your hard work and your newfound financial prowess. You've proven that with a little knowledge and a lot of determination, you can conquer any financial puzzle that comes your way. Now, go forth and invest wisely!