 We often say we evaluate a function or a formula and solve for a given variable. But the only real difference is that a function or formula is already solved for one of the variables. And in general, given any formula, you should be prepared to solve for any of the quantities it uses. So, for example, our simple interest formula has four quantities. The amount a k t at some time t, the principal k, the interest rate s, and the time t. We can use the formula to calculate an amount and solve for any of the other quantities. So, for example, suppose a friend borrows $20 and pays you back $25 a week later. What kind of person are you that charges a friend interest? Wait, wait, moral and ethical questions are not part of mathematics, so let's just consider the mathematical question about the interest rate per day. Now, since we're looking for the interest rate per day, our time t must be measured in days. And so we have the amount borrowed, the principal is 20, and the amount of time is 7. And since they're paying us back $25 after t equals 7 days, then a 27 is equal to 25. Now, equals means replaceable, so we know the value of a 27 k and t, so we'll replace those. And to solve an equation, the thing to keep in mind is the last thing you do is the first thing you undo. So, over here on the right, we're multiplying by 20. And since the right-hand side is a product, we'll begin by dividing both sides by 20. Then we'll subtract 1 and then divide by 7 to get s, our interest rate per day, which we'll convert into a percentage by multiplying by 100 and then rounding to 3.6%. And at this point, it's important to consider rounding. Now, in physics, engineering or chemistry, a rounding error might cause minor consequences like a bridge collapsing or people dying, but in finance, a rounding error can cause people to lose money, so it's important to worry about rounding. And so here's an important idea. Never use a rounded value in any computation. And in particular, only round the number you report. In other words, once you've rounded, don't ever use that number again in anything. Now, the best approach here is to do all your computations at the same time, but that can be a little bit cumbersome. The second best approach is to use the capability of your computing device to store intermediate calculations. And most calculators and spreadsheets are designed to do this, so take advantage of your technology. So if we wanted to solve 25 equals 20 times 1 plus 7 s, we'd go through the same steps. We divide by 20. We subtract 1 and divide the whole thing by 7. And here we've recorded what our computation should be, but we don't actually do them until we calculate at the end of the problem this single expression. Now, if you plan to take something like the actuarial exam, a certain amount of technology is allowable, and it's important for you to learn to use the available and allowable technology. And in particular, I'm not going to talk about that because of two reasons. First of all, some of you already know how to use it, and second, that technology changes constantly, so you will lead to learn how to use technology on your own at some point. And the best place to learn technology on your own is in the classroom where, one, you can determine whether or not you've used it correctly by comparing your answer with the right answer, and two, the worst thing that will happen if you don't get it right is you'll get a bad grade. In contrast, once you're out in the real world, if you don't learn how to use technology correctly, your company will find someone who can. In any case, we want to evaluate this expression to find the interest rate, so we can use a calculator, and most modern scientific calculators will actually allow you to input something that looks exactly like standard math notation, for example this, and so we find our interest rate 0.03571. So, another quantity in our simple interest formula that we could solve for is the amount of time. So I suppose you invest $1,000 at 7% simple interest annually, how long before the investment is worth $2,000? From our simple interest formula, akt equals k times 1 plus st, we have the principal amount $1,000, we have the interest rate 7%, that's 0.07, and we want a 1,000t to be 2,000. We want to know when our investment will be worth $2,000, and we don't know t, so we can write down our formula and solve. And again, keeping all of our calculations until the end, we find t is approximately 14.29 years. And just this day I got an email from a Russian nobleman who says he's willing to pay 45% simple interest annually. How much should we invest to make $100,000 in two years? So, from our simple interest formula, we have our interest rate 0.45, our time 2, and after two years we want the amount to be $100,000, with k the principal amount as our unknown. So we can write and solve. And so this means you should invest probably nothing because something that sounds too good to be true probably is. But it's your money.