 In this presentation, we will take a look at how to calculate simple interest a few different ways. As we look at this, you may ask yourself, why are we going over this a few different ways? Why not just go over it one way, the best way, and let us learn that well and be able to apply it in each situation? While one way does work in most situations, in other words, we will probably learn one way have a favorite way to calculate the simple interest and apply that in every circumstance, it's also the case that when we look at other people's calculations or a technical calculation, they may have some different form of the calculation. For example, I prefer a way when I think about the calculation of simple interest to have some subtotals in the calculation and have more of a vertical type of calculation the way we would see if done in something like a calculator. If we see a type of equation in a book, then the idea there is to have the most simple type of equation expressed in as short a way as possible. And that typically is going to be some type of formula. And that formula will often not be showing the subtotal. So in other words, a textbook has an incentive to show a very compressed type of format for calculating something. And individuals, if we want to go back to it, note what happened in it, then we often will benefit by having some subtotals in the calculations that we work through. That's one type of difference we want to know. Other reason we want to know different methods is just to understand the math a bit better. If we understand different approaches, we understand what we are actually doing a little bit better. Another reason is that different people are going to have different minds in terms of how they think of things and how they process things. So we want to be able to look at someone else's work and say, okay, I see what they did here and be able to apply that to ourselves and see what they're doing. And again, the more we can understand how other people process things, the better we understand what the material is and how to communicate it to other people. So let's go through a few of these type of ways we can calculate simple interest. So we're going to start with a loan of 50,007% interest rate and it's a 90-day loan. It's going to be out for 90 days. So we'll start out with the loan amount, 50,000 for our first calculation type. We're going to multiply that times 7%. Now, note if we have a calculator, we're typically going to do that by saying 50,000, 50,000 times 0.07. 0.07 if we're in Excel, we format these Excel to be percent. We can have a percent. We can use this percent button up here, but I typically use decimal 0.07. If we move the decimal two places to the right, then of course, we would have a 7% and that would give us the 3,500. So in other words, 50,000 times 7% is 3,500. First thing we need to note, this 7% means 7% a year. Unless expressed otherwise, whenever we say that something is so much percent, typically we mean percent a year. So for example, if we say we have a mortgage, we pay the mortgage monthly, but we typically express the mortgage rate as a yearly rate. Why would we do that? There's a couple different reasons we might do that, but one main reason that it's a convention for us to do that is if we took 0.07 and said we wanted a monthly type of interest rate and we divide that by 12, we get a pretty small number. So it would be 0.58 something, something percent, which is pretty small. We don't really, it's a range that's more difficult for us to express in. Therefore, we typically express interest rates for that reason and just it's a convention in years. So unless expressed otherwise, just note within a problem within any type of discussion, interest typically means a year. Similar to if we're talking about a salary, like we say someone makes $70,000, we typically probably mean $70,000 a year, even if not stated. Then we're going to take that amount and we're going to say how many days are in a year. Now in this calculation, we're taking $360,000 to make an even rather than $365, which is an assumption of 12 months times 30 days in a month. Of course, there are 31 days or 30 days or 28 days or 29 days in a month. But if we do a nice even calculation here, we get an even 360 for our simple interest calculation. So that's what we will use at this time. So then we're going to take the $3,500 per year of interest. This is a dollar amount. And this is going to be 360 days, gives us about $10 a day. Again, that's rounded. It's really 3,500 divided by 360 gives us 9.72222. So I'm going to use this 10 here just to show it. That's how you'll see it in Excel if you take the decimals off. But note that we're really going to be multiplying by that 9 something. And just remember when you deal with interest, you're going to have to deal with that. Not everything's going to run out to the dollar. Not everything's going to run out to the penny. We'll have some rounding differences. So then we're going to take that number times 90 days and that'll give us 875. Again, not 90 times 10 in this case because we're using the rounded numbers as we would see in something like an Excel worksheet. So you just got to kind of deal with those kind of things. So if it was 10 times 90, it would be 900. What we really have is the 360 divided by 3,500. Let's do that one more time. What we really have is the 3,500 divided by 360 or 9.72, which was rounded to 10, taken off the pennies. And then we'll take that and we multiply it times 90 days. And that gives us our 875. So then if we take our 875, that's how much we would earn then for the time period for the 90 days at 7% for simple interest. If we take the original amount of the loaned 50,000 we loaned out at day one, we add to it the 875, then we get 50,000 875 that we would then receive at the end of the loan term. This is the way that one of the ways I think it's the most easy to see this. And that's if we break it out by days. It depends on the loan type if we want to break it out by days or months. So if it's less than a year, then it might be that we want to use days. If a problem is getting very specific, then we'll have to use the actual number of days in the month. Just got to be careful on the terms of the problem. If it's going to be a longer term problem, then we might just round these two months and use months as the calculation as we'll see this time. Same type of idea. We'll take the 50,000, multiply it times 7%, that's going to give us our 3,500. So 50,000 times 7%, the yearly rate gives us the 3,500 interest per year. Now we're going to take that and we're going to, instead of dividing it by 360 days, we'll divide it by 12, giving us the interest rate per month, 3,500 divided by 12. So again, if we did that with the calculator, we're taking 35 divided by 12. That gives us not exactly 292, but 291.6666 on forever. Note that even if I put the pennies there, it's not always going to be exact. And we're just going to have to deal with that. That's going to be some rounding issues we will always have when we do these types of calculations. Then we're going to take that amount and multiply it times 3. Where do we get 3? 3 months, 90 days divided by our convention 30 days per month about gives us 3 months. Then we're going to take this dollar amount per month times the number of months, 3, to give us that 875. Remember that we're not talking about the 292 necessarily, we'll have rounding there. It's an off by dollar because really what we're talking about is the 3,500 divided by 12 or 291.6666 on forever times 3. And that'll give us our 875. So just be careful of the rounding. It's always going to be a problem. It doesn't matter how many places we take it out to. Some of the decimals will repeat forever. So we'll have the 50,000 plus the 875 getting us to that same 50,875. These two ways, the prior way by day and this way by month are the easiest way for me to think about it because it's linear. This is how you plug it into a calculator. This shows us each step and the subtotals. However, if you were to represent it with a short way as possible, then these subtotals aren't something that are typically going to be represented in textbooks because it's not as neat, not as nice. So when you put things out and you calculate them, I would remember them in this format. This is easier for me to remember than a shorter type of formula because the shorter formula doesn't make intuitive sense. I can't say, well, walk through this and say, well, this is the yearly interest rate. This is the rate per year. I mean, and then we're going to take the rate per month times the number of months to get the interest. So I can't really tell the story to myself. The longer story that I can tell is easier for me to remember than a shorter formula, which has no story. If that makes any sense. So anyways, bottom line, I would remember one of these two methods for your default method. Okay, so now we're going to do another method. We're going to start off with the number of days in the loan 90. We're going to divide that by the number of days in the year. This of course is called a ratio. So we're comparing and starting off with the comparison of the loan term 90 days divided by the number of days in the year 360, which again is 12 times 30 for our purposes, rounding it instead of 365 is 360. And that's going to give us 0.25 or 25%. So here's our ratio given us 0.25. Now that we have that ratio, we can take that ratio that 0.25 and multiply it times the interest per year, which was that 3,500 or 50,000 times 0.07 that 3,500. And that'll give us our 875. So note this gets that 875 a bit faster here. We don't have as many calculations or as many steps. However, I find that most students and myself included, when I think about this, the ratio doesn't make as intuitive sense to me as saying, oh, here's the interest rate per year. Here's what the interest would be per year and then breaking it out to either a daily or monthly interest rate and then multiply times the number of months. Same math, same end result, just a different order of calculations. So just be aware that this, because it's smaller, probably often more represented in a textbook or something, we need to be able to see that. So then we got the 875 plus the 50,000 original gives us the loan that we'll get back at the end of this time period. If we loaned out 50,000 at 7%, simple interest 90 days, we should get back 50,875. Next, we'll do the same type of thing. But now we're going to do this by the number of months in the loan. So remember this is going to be 90 divided by 30. Gives us three months in the loan. And then we're going to take that and compare it to 12. 3 divided by 12, 3 over 12. That gives us the ratio of 3 to 12. So 3 divided by 12 will give us that same 25. So note we're doing the same thing, of course, because we're doing the comparison of the time frame. The loan is over over the total time frame. And as long as we use the same measuring tools, long as we're in this case using months, or in the prior case, we used days for both the loan term and the amount of time within the year, we will get to the same ratio 0.25 is the percent. So then we're going to, and of course, you can simplify these down to and see that the same ratio of 3, 3 over 12, if you simplify that down, then you simplify the 30 compared to, or the 90 compared to the 360, you'll get to the same ratios, right? So there's the 0.25. And then we're going to take that and multiply times the 3,500, which once again is, of course, the $50,000 loan we started with times 0. Times 0.07, 0.07, 3,500, and that'll give us the 875. So 0.25 times 3,500 gets us back to that 875. So once again, it's quicker method than the first two we started off with, a little less intuitive in my experience with myself and people I've worked with for most people understand, I believe. Total cash received then would be this $50,875. Next method. Now we're going to try to break this percentage out as we saw at the beginning and break it out into a monthly rate. And we'll deal with that. So we're going to say we're going to start with the interest rate this time and say that's the yearly rate. That's what default by default typically is divided by 12. And that'll give us the monthly rate. So once again, if I did this one, a calculator, we'd say 0.07, that's the 7%, divided by the number of months in a year, 12, gives us 0.0058333 on forever. That's a pretty small number, which once again is the reason we don't typically do that. We don't typically represent the interest rate in terms of months, because it typically is a small number, and it's just not the convention that we just have out of whatever reason we come up with conventions for. So that's what we're going to have if we break it down to a monthly rate. Now it's important to know this, because when you put stuff into Excel, or other types of financial calculators at times, then it sometimes uses this monthly rate. So we'll need to know the rate per period, and that's in this case the monthly rate. And so it's important for us to know this type of calculation when we use tools such as Excel. So that'll give us 0.58%. Remember that is not exact here. This is a rounding times the loan of $50,000, gives us $292,000. So once again, be careful of the rounding all the time. Because this is 0.598, you might first think it's 58%. It's not. We got to move the decimal two places to the left, which was 0.0058, is what it really represents, times 50,000. But even now it's a couple bucks off. Why? Because this is rounded too. So remember what we had? It's 0.07 divided by 12, gives us 0.00583333, and that's what we're using times 50,000. So again, just be careful. I could give an example that is perfectly even here or but it's not, I'd rather give a less than perfect example to show that it's not going to be perfect all the time. It'll be perfect sometimes. So this would be the amount of interest per month, simple interest per month. And now we just need to say that's a dollar sign, by the way. Now they're going to say that there's how many months in the year, we're going to say that in the term of the loan three, or 90 divided by 33 months. So 292 times three would be 875. Once again, that might be a bit of rounding there. If we say 292 times eight times three, we're going to get 876. Why? Because really what we had, remember is 0.07 divided by 12, which is 0.0058. And then we multiply that times the 50,000 to get really 291.6666 divided our times three. That's going to be our 875. And then if we take the 875 plus the 50,000 gives us that same 50,875. One more time, one more way, and then we'll stop this. Okay, so we're going to take that same interest, 7%, break it out to a monthly rate. So 7% divided by 12 gives us that 0.87, 0.58%. Once again, this is rounded, same spot we were at last time, we now have a monthly percent percent. Now we're going to take that and multiply times the number of months. So we have 90 days divided by three divided by 30 gives us three months. So if we take that 0.58 rounded, this is rounded times the three, we get about 1.75. And this then would be the interest rate per the time period we're talking about three months or 90 days. So in other words, we take a look at this, we have the interest rates 0.07 for a year divided by 12 gives us the interest rate per month by moving the decimal point over 0.58333. This is rounded. This isn't 58%, remember 0.58 big difference. And then we're going to take that times the number of months three. And that's going to give us 0.0175 or 1.75%. So this is actually exact now 1.75%. And then we're going to take that times the loan amount 50,000. And that'll give us the 875. So remember this, what we did here is we broke down the percent for a three month time period or 90 days. And again, that's not usual. We don't usually say that. We don't usually say, Hey, we're going to pay you a simple interest rate of whatever for a three month time period. So just, you know, be aware that we don't typically say that we're going to pay you 0.58 per month. We typically express things oftentimes with a yearly rate that yearly rates often being between one and like 20. And therefore being an interest rate that makes more sense to use within that range. So, but when we do financial calculations, this once again might be a way that we could see this within the periods that we're talking about, which in this case, is a 90 day or three month period. So then we just take that interest rate times the 50,000 gives us the 875. We add that to the original 50. We get to that same 50,875. So again, I hope this wasn't too confusing or more confusing than it was helpful. But note that when you see this type of calculations in a textbook, they're often so simplified, they look to be like the easiest way to see it. And they are the easiest way to write it. But obviously, oftentimes a longer type of calculation, a more vertical calculation rather than a horizontal type of formula is easier for us to think through, tell a story in our head, remind ourselves what we're actually doing in order to better memorize the calculation for future use in application to other types of problems, as well as to just have a format and be able to understand it better for future use and know what we're actually doing rather than spitting out a number that we think is correct just based on the formula, but not having any intuitive realization or understanding what it's for and therefore how we can use it to apply to something.