I am Chloe. Now that we have seen the visual “Growth Race” and we know that compound interest is an example of exponential growth, we need to take the next step. It’s time to move beyond tables and graphs and learn how to calculate these values exactly using the master mathematical formula. This formula is the key to predicting your future wealth, and it is a core part of the MCF-3-M and MCR-3-U curriculum in Ontario. We are going to break down the formula $A$ equals $P$ times one plus $i$ to the power of $n$. It might look a little intimidating at first because of that exponent, but I promise, once you see what each letter represents, it’s as simple as following a recipe. Let’s get to the lab and break it down!I am Liam. I know that math formulas can sometimes feel a bit overwhelming, especially when you start dealing with decimals and exponents. But that is why we have technology! I will walk you through exactly how to use your scientific calculator so you never get stuck. We will look at the specific order of operations and the special buttons you need to find. Whether you are using a physical calculator or an app on your phone, the logic is exactly the same. We’re going to make sure you can punch in the numbers and get the right answer every single time, whether you are calculating the future value of a savings account or the cost of a student loan. Let’s make the calculator work for us!I am Maya. While the big formula gives you the final amount of money in an account, there is one more thing you often need to know: how much interest did you actually earn? Or, if it’s a loan, how much extra did you have to pay? I will show you how to find the total interest earned by comparing your final amount to what you started with. We call this the “Interest Formula,” and it’s a simple but vital subtraction step. It’s the best way to see the actual “cost” or “profit” of your financial decisions. I’ll also give you some tips for your tests to make sure you don’t accidentally give the wrong answer when a question asks for “interest” instead of “total amount.”And I am Noah. This formula isn’t just for looking forward into the future. It’s also for planning today. We’re going to talk about something called “Present Value.” If you know you need to have a certain amount of money in three or four years—maybe for college or a new car—how much do you need to invest right now to reach that goal? This is just the compound interest formula worked backward. We’ll look at the variables and even show you a neat trick using negative exponents. This is the secret to real financial planning. By the end of this video, you will have the “Formula for Success” in your pocket, ready to use for any financial challenge.Let’s look at the variables in our master formula: $A = P(1+i)^n$. First, the letter $A$. $A$ stands for the Final Amount, which is also sometimes called the Future Value. This is the total amount of money you have at the end of your investment. Next is $P$, which stands for the Principal. This is the starting amount—the money you put in on day one. Then we have the small letter $i$. This represents the interest rate per compounding period. This is a very important detail: if your annual interest rate is five percent, you must write it as a decimal, which is zero point zero five. Finally, we have $n$, which is the number of compounding periods, usually measured in years for school problems. Because $n$ is an exponent, it means the growth isn’t just adding; it’s multiplying over and over. This is the exact definition of an exponential function.Let’s try a real example together so you can see how this works on your calculator. Suppose you have five hundred dollars—that’s our $P$. You put it into a high-interest savings account that pays five percent interest—that’s our $i$—compounded annually. You want to know how much you will have in three years—so $n$ is three. First, we set up our equation: $A$ equals five hundred times, open parenthesis, one plus zero point zero five, close parenthesis, to the power of three. Using your calculator, you always start inside the parentheses: one plus zero point zero five equals one point zero five. Now, here is the secret: find your exponent key. It usually looks like a little hat or the letters $x$ to the power of $y$. Raise one point zero five to the power of three. This gives you about one point one five seven six. Finally, multiply that by your five hundred dollar principal. The total amount, $A$, is five hundred and seventy-eight dollars and eighty-one cents. See? It’s just a few button presses!Now that we have our final amount, $A$, of five hundred and seventy-eight dollars and eighty-one cents, let’s find the interest. We use the formula $I = A – P$. In our example, we take our total amount and subtract our original five hundred dollars. This leaves us with seventy-eight dollars and eighty-one cents in interest. This is the “profit” you made just by letting your money sit in the bank! A huge tip for your tests: always read the question carefully. If it asks “How much will the investment be worth?”, give the $A$ value. If it asks “How much interest was earned?”, you must do this subtraction step. Also, a quick sanity check: in compound interest, your final amount $A$ should always be larger than your principal $P$. If your answer is smaller, you probably made a mistake with your decimal or your exponent. Compound interest is about growth, so the numbers should always go up!Sometimes, the question is reversed. This is what we call “Present Value.” Imagine you want to have one thousand dollars in five years to go on a graduation trip. If the bank offers you a four percent interest rate, how much do you need to deposit today? To find this, we use the formula $P = A(1+i)^{-n}$. Using a negative exponent is a brilliant math trick that tells the calculator to “divide” or “go backward” in time. If you punch in one thousand times, open parenthesis, one point zero four, close parenthesis, to the power of negative five, you get eight hundred and twenty-one dollars and ninety-three cents. That is your present value! Whether you are looking forward or backward, the relationship remains exponential. The pros of mastering this math are that you can compare different bank offers and see exactly how much your money will be worth. Motivation comes from seeing those numbers grow and knowing exactly how to reach your goals. In our next video, we will look at how this same math works against you when you have debt and credit cards.
