 In this presentation, we will cover a few different ways that we can calculate simple interest. As we look at these different ways, you may ask yourself, why would we want to know different ways to calculate simple interest? Why not just pick one, the best one, the one we like best, and use that at all times? You probably will have a default method that you'll use at all times. But if you look at different areas, different people's calculations, or different types of calculations, or different terms of a loan or a textbook answer, they may put it in a different format and we need to basically be able to understand what format they have it in and why it is that way or how to calculate interest or how to look up what has been done. All of these are going to be variations on the same math, however, of course. We're just putting the order of operations in different ways in essence when we look at these different types of methods to calculate the same end result. So first, for example, we'll say that there's a $50,000 loan, 7%, it's going to be simple interest. We're not dealing with compounding. We're just going to talk about how to calculate interest, the simple interest for 90 days in this case. 90 days would be about three months. Note that you could have loans that are going to be termed in terms of years, terms of days, terms of months. Most type of book problems will try to make the months kind of work out evenly and most a lot of book problems will use months that will be assumed to be similar in the number of days or the same in the number of days, 30 days in other words per month and those are some assumptions we'll make here. So that means there's 360 days in a year, 12 times 30, if we make the assumption that all the months have 30 days rather than 31 and 28 and 30, we'll just make the assumption they all have 30. So those are some of the things that we want to keep in mind as we work through these problems. Okay, so first let's just calculate this a couple different ways. We're going to say first type of calculations, we have the principal of $50,000, we're going to type that in here. The interest rate we're going to say is 7%. So here because the cell is formatted, I can just type in 7 and it'll give 7%. Remember that if we typed that into our calculator, it would be equivalent to 0.07, 7%, move the decimal places two places to the right. Within Excel that's in the home tab, align our numbers group, and you could make it a dollar, a currency type of cell, now that I'm on this cell, we could make it a currency type of cell, which is 0.07, and then it just moves that decimal point two places over, adds a percentage sign when we select the percent. Then we'll multiply this out. So I'm going to do this with a formula. This equals 50,000 times the 7% B4 times B5, and once again if we had the calculator here, we're just going to say that is 50,000 times 0.07, 3,500. It's important to note that this 7%, whenever we say a percent unless it's stated otherwise, we mean for a year. That's the default terminology for a percent. Percent means per year. If we talk about a mortgage, we pay it monthly. We don't give the percent in a monthly format, typically. We say this is the rate of interest per year. And then we need to take that and break it down. You might ask why we do that. One reason is because percentage that would be broken down to a monthly percent would be pretty small. We'd be talking about fractions of numbers, fractions of percents, whereas a year, then we have percents that typically lie between 1 and 20, which is a good range for percentages to be explained in. So that's one reason, but just that's the convention. So if we say that there's 3,500, then if this loan was out for an entire year, however, it's only out for 90 days. So we need to break this out to 90 days. One way to do that is to think of, well, let's break this out from a yearly total to a daily total and then multiply times the 90 days. So there's 360 days in a year if we assume they're all 30 months. There's really about 365, but we're going to make this even by saying 12 months times 30 days per month average. That's nice even, 360. So if we take our percent per year, the dollar amount per year, and then we divide by 360, we're going to say this equals 3,500 divided by 360, and that gives us $10 a day. So $10 a day would be the interest per day. And now we're going to have the number of days in the loan. So this is per day, another 60 days in the loan. So we're going to 90 days, sorry, 90 days in the loan. So we have the 90, so 10 times 90 then, we're going to say this equals 10 times 90, and that gives us 875. So this is the amount of interest then, simple interest that would be earned at a 90-day loan, 7%. That means that if we loaned out the money, had a simple interest to be paid back at the end of the loan, we'd get 50,000 back plus the 8,875, or this equals the 50,000 plus the 8,75. Now this is one of the most intuitive methods to me to write this out, although it is a bit longer. We are doing step-by-step method. Notice what we're not doing is writing it into an algebraic formula and trying to put it all on one line, which we can then do in Excel with order of operations because a lot of times it's easier to just write it out like this, which is how you would in essence see it in a calculator. If we did this in a calculator it would be 50,000 times 0.07, that would be 3,500. For a year's worth of interest, we would divide that by 360, and that gives us around 10. Notice this isn't exact because there's no interest here, I mean there's no decimals, so if I select this cell, home tab, numbers, and increase the number of decimals, it's really 9.72. I'm rounding it to around 10, but because we're using the cell in Excel, because we're multiplying 10 times 90, it's using the actual number here, not 10, because obviously if we said this equals 10 times 90, we would get 900. That's not what we get here, so just use Excel and note that that's a benefit in Excel, and Excel gives us the exact number, even though it's not showing all the decimals here, so you have to just be aware of that. In this case we're back here, and we have this number, and then that would be this subtotal, and then we multiply times 90, and that'll give us our 8.75, and then we add the 50,000 original to it. This way that we have the subtotals in the calculator, one calculation at a time, representing it from a vertical top-down approach, is often easier to see in actually writing in the subtotals, other than writing in an algebraic equation, just a linear equation that doesn't have the subtotal. This is the way I think it's most simple for me to see it, and for students I've seen learning this, I think they pick it up most easily when we actually lay it out in this fashion. We can do the similar fashion, similar type of calculation, but considering the fact that the 90 days is equivalent to three months, if, you know, 30, 60, 90, three months, 90 divided by 30, three months, if we consider each month having an equivalent amount of 30 days, which we'll do for simplicity for simple interest calculations here. So same type of idea then, we have the 50,000, 7%, and we're going to multiply that out. This equals the 50,000 times 7%. Note, once again, the 7% means 7% a year. So $50,000 loan times 7% a year gives us 3,500 for total interest if the loan was out for a year, which it's not, it's only out for 90 days. So then we need to say, I'm going to break this out instead of two days, 360, we'll break it out two months. So we're going to say how much per month in interest would be earned. So to do that, we'll take this divided by 12 and we'll say that this equals the 3,500 divided by 12 and that will give us 292. Again, this may not be exact, home tab numbers increase the decimals. It's actually 291.67. However, if we use Excel, it'll do the rounding for us. So then we're going to go here and we'll say how many days, I mean how many months are in the loan, 90 days. So I'm going to say this equals 90 divided by 30. 90 days divided by 30 means about three months. So if we multiply the 292 times three months, we're going to say this equals 292 interest per month times the number of months, three, giving us 875, matching what we got before. Once again, if we add that to the principle, then we're going to say this is the original amount of the loan. If we loaned that out at 7% or 3 months or 90 days, simple interest, we should get the 50,000 plus the 875 or 50,875 at the end of the term. Okay, another way we can do this, we're going to take this same thing. We're going to say the number of days in the loan term and this will be kind of a more of a ratio type of approach here. So we're going to say we're going to first start with the ratio. So the number of days in the loan term is 90. 90 days in the loan, number of days in the year is going to be 360, which is the 12 times 30. And we're going to take this ratio, the 90 compared to the 360, the 90 over 360 or 90 divided by 360. This equals 90 divided by 360. That gives us our percentage or ratio. And again, it could be a rounded home tab number. If it were, you could see the decimals here. It's not this case. So it's 25%. And then we're going to take that and times it by the interest per year, which we've calculated a couple times is that 3,700. So I won't do that. I'll just do it in one calculation here. It's the 50,000 times 0.07. That's the 3,500 per year times the ratio. And note, that's what we did up here, same type of thing. Sorry, up here. Sorry for going back and forth like that. But we had the 3,500 divided by 360 times the 90. So instead of we're just reversing the math, we're just doing a different order of the algebra. So now we're just taking the 90 over the 360, giving us the ratio or the percent of 0.25, 25%. And then we're going to multiply that times the interest per year. So we're going to say this equals that 0.25 times 3,500 or 875, that same 875 as here. And that means, of course, that if we take the original 50,000 plus that 875, we should once again get to that same 50,875. We can do that same kind of ratio. Notice, we did it here with days. We can also say, let's take the number of months in the loan term. So if there's 90 days, so this equals 90 divided by the number of days in a month, we're going to say 30 on average means there's three months in the loan compared to the total months in a year, 12. So the ratio now being 3 over 12. We're going to do that to math here. This equals 3 divided by 12. That gives us the 0.25, 25%. And then we'll take that times the interest per year, which once again equals the 50,000 original loan times 0.07, 7%. So there's our 3,005. So we're at the same point now as our prior calculation doing the ratio not with days, but with month, same ratio. This equals the 0.25 times the 3,005. There's our 875. And therefore, the cash back at the end of the loan term would be the original 50,000 we loaned out plus the 875 or 50,875. Another way we can do this, we're going to take a look this time at dividing out and getting the interest rate per month first. So we're actually going to change the interest rate from, as we've already said, a yearly rate and look what would happen, what if we changed the rate to a monthly rate. So let's do that. We're going to say the interest rate per year is 7%. And we're going to say the number of months in a year is 12. So the interest rate per month is simply going to be equal to the 7% divided by 12. Okay, now if we did that in a calculator, it would just be 0.07 divided by 12. So it's 0.00, I mean 0.0058. And then if we move the decimal 0.2 places over 0.58 and it's actually 0.58333. Note how small of a number that is, which is why one reason for us not representing interest in terms of a monthly rate is too small typically. So note if I go to the home tab and make it more decimals, make it more decimal, you can see it's really 0.58333 on forever. Excel sees that, recognizes that even though it's rounding it to two places as we've determined here as we've told it to do. Then we're going to say the loan principal amount was the 50,000. So if we multiply the principal times the rate per month now, we're going to say this equals the 0.58 times 50,000, giving us the 292. Once again that could be rounded, home tab, numbers, it's rounded. But we're going to use that figure and then we're going to take the 50,000, sorry, number of months in the loan, which is 90 days. So it equals 90 divided by 30 or three months. And we'll take the 292 per month times three months, this equals 292 per month times three months gives us that same 875. Once again, if we have the principal equal to 50,000 plus that 875 at the end of the loan, we will get 50,875. Next one, we're going to say this one more time similar type of calculation, we're going to start with the interest rate per year, which is going to be that 7%. We'll say 7%. Number of months in a year are 12. And then if we do the division here, we're going to say this equals that 7% divided by 12, giving that same 0.58. Once again, remember, if I go to the home tab, numbers, add some decimals, it's really 0.58333. Taking those back off, number of months in the year. So now we're going to multiply this times the number of months in the loan, number of months in the loan, I changed the term there a bit should be the months in a loan. So this is going to equal the 90 divided by 30 or three. So we're going to take this is the interest rate per month. And if we multiply times three, we're going to say this the interest rate per month times three, then this would be the interest rate per three months. So then we're going to say home tab numbers. And that's rounded it's even at 1.75% per three months. And then if we take the loan term 50,000, now we have the interest rate per what the loan term is three months. And we can just do the multiplication here. We're going to say this equals the 1.75 times the 50,000 gives us that same 875 once again. Then we're going to say this equals the 50,000 plus the 875. So those are just a couple ways that we can do this. Again, I think the easiest way is one of these two up top. These are the ways I like to see it. This is the interest per year. This would be the interest that would be provided per year and then break it down to per day. However, this method down here of finding the interest rate per month is very common when we use Excel because it's often part of the formulas, meaning we typically oftentimes when doing time value of money, take the interest rate and divide it by 12 or find the rate per period, which is going to be important. So it's important to know that. And also, again, you can see this many books will just represent it's different ways, and there's a difference between wanting to represent it in terms of the shortest way to do it, which would be good for a textbook or something like that to represent the simplest formula and representing the best way to think through it from a human being trying to understand it type of position. And that's where a longer type of subtotal type calculation, I believe is better. So notice there's a conflict there. If you want to see something as concise and as nice and small as possible with the fewest factors involved to get to the same result, then you're going to use probably a more simplified formula. If you want to understand it and be able to break it out in your mind and be able to go back to it and say, oh, I see what happened and why we're doing it, then you're probably going to have more subtotals or a vertical type of fashion when doing these type of calculations.