 In this video, we're gonna introduce a very important type of linear model referred to as simple interest. And so familiarize you to some of the vocabulary associated here, think of the following situation. Suppose a loan is given out to a daughter who's fallen hard times by her parents, right? You know, parents wanna kinda help her out here. The parents, they wanna give their kid the money, but they don't just wanna give it as a gift, they wanna learn a little bit more responsibility, and therefore they're gonna give her a loan. It'll be a simple interest loan for which there is interest collected, but it's not as unnecessarily malicious as maybe it would be from a bank or something like that. It's a very generous loan here. And so simple interest is based upon the following idea. So first of all, she's gonna borrow a certain amount of money, right? And this is called the principle. This is how much for the borrower, this is how much money you take away. And for the lender, this is how much money you're giving. Now, the idea of a loan is that this principle will be paid back to the lender from the borrower, but it'll be paid back with some extra interest. So some percentage of the original principle we've added back to it. And so let's say that the interest rate is gonna be our percent in the situation. And this is gonna be our percent per some unit of time. Most financial loans are done in a year timeframe, but if it's a short-term loan, it could be a matter of months or weeks. But so yeah, typically this would be like your percentage per year. So you talk about like your APR, your annual percentage rate, it talks about an interest rate per year. So after an agreed upon timeframe, the daughter must repay her parents the original principle plus interest on the principle that occurs while she's in possession of the loan. So if she repays the loan back after T units of time, she will have to pay back her parents the following things. So you get this formula right here. So the amount that she has to pay back her parents is gonna be P, the principle times one plus RT, which with this simple interest loan, there's not any recurring payment. It's just gonna be a lump sum. Let's say that she borrows $6,000 from her parents, then let's say like six months later, three years later, 10 years later, she gives back all the cash to her parents. She'll give back the original principle and interest all at once. Where this formula comes from is the following. If you distribute this, you're gonna get P times one, which is just one, right? She pays back the original principle plus P times R times T. So what PRT here represents is P times R is a percentage of the principle. If she has to pay back 10% of her $6,000 loan, she's gonna be paying $600, but $600 per unit of time, right? So $600 per year, maybe 10% per year. So every year he takes to pay back, she has to pay another $600 back. And so when you factor out the P, you get the typical simple interest formula, the amount equals P times one plus RT. Now, when you distribute it out again, right? Going back to the distributed amount, you're gonna get P times R times T plus P, in which case, what you see going on here is that this is actually a linear function where with the linear function, your variable is time. How long does it take for you to pay back the loan? Your Y-intercept is the original principle if you borrowed the money and then paid it back immediately, no time elapsed, then you would pay no interest and you would pay just back the principle, that's the Y-intercept. And then the slope here is gonna be the interest rate applied to the principle, right? P times R, how many dollars per year do you have to pay back for the loan? And so simple interest is a linear model, even though it's generally written in this factored form. So suppose the daughter borrows $6,000 and she agrees to pay 6% annual percentage rate APR on the loan and she pays back her parents after three years. How much money does she owe her parents? Well, in that situation, using the simple interest formula from above, the amount will equal the principle, which is 6,000, times one plus the interest rate, which is 0.06, make sure you write your R actually as a percentage, length is a decimal, not with the percentage symbol and then she pays it back after three years. This percentage rate is percentage per year. And so since this is year, there's no conversion of time we have to worry about. This is the calculation we do. So we're gonna first do 6% times three. That's actually gonna have the effect of 18%. So taking three years to pay back 6% per year is the same thing as having 8% interest rate after one year. You add that together. So you get 1.18 here and then multiply that by 6,000. That gives us 7,080. And so that's how much the daughter would have to pay back her parents after three years. Be aware that if you subtract 6,000 from that, that means she's paying $1,080 in interest. That's the cost of the loan, which of course, this is just gonna be PRT, which oftentimes when people use the interest, how about simple interest? They sometimes write it this way, that interest is equal to PRT. That just gives you the interest. You have to add the interest to the principal to tell you how much you'll pay back at the end. That's an important thing. But the interest itself, this is how much it would cost her for the loan. Let's switch up the situation again. Let's suppose that the daughter borrowed 6,000 and she has an APR of 6%. But let's say that she pays back her parents after a timeframe of five months or something like that, right? Let's say she pays it back after five months. Now, when you are working with months, right? You'll see that the month, there's a mismatch between months and percentage rate because the APR is an annual percentage rate. It's per year, right? This is percent per year, but we're measuring time in months. So we have to convert from one from the other. We could switch five months into years, in which case we'd be five-twelfth of a year. That's perfectly acceptable. But we also, we could switch percentage per year. We could also switch this to percentage per month by taking percentage per 12 months, like so. So basically, we have to either take the timeframe and divide it by 12 so that it becomes years or we have to take the APR and divide it by 12 to make it a monthly percentage rate as opposed to an annual percentage rate. For simple interest, it doesn't make really much of a difference which direction you go here. The amount that she's gonna have to pay back is 6,000 times the principal. Then you're gonna get one plus, we're gonna get R, right, is the next number. I'm gonna write the percentage rate as percentage per month. So we're gonna get 0.06 divided by 12. So that gives you the monthly percentage rate. And then we times that by five months, like so. And so now we just have to multiply these things together. Let's see. So when we do that, 0.06 divided by 12, this is going to give us, let me write it out here. This is gonna give you 0.005, you times that by five. We then get one plus, you're gonna get 0.0025. So it's a zero right there, I promise. Add that to the one, you have 1.025. And then times that by 6,000, that would end up with 6,150. That's how much money she'd have to pay back if she paid her parents back after five months. And so if you focus just on the interest part, right, the interest is just gonna be $150. So it's a dramatically smaller interest that she has to pay, not because the interest rate changed, not because the principal changed, but because she paid it back faster than it's less interest overall. So it only cost her $150 to borrow it for five months as opposed to borrowing for three years. So if anything, you get out of the lesson like this is if ever you borrow money, the key to financial success is to pay back your loan quickly. The longer you take, the worse it'll be for you. And that might mean making sacrifices. If you graduate from school and you have some student debt, the key I want you to know here is pay off your debt as fast as you can. That might mean you can't buy the Xbox 17 that just came out for another year or two. Show a little bit of self-discipline and pay off your debt because you'll have actually more money in the future if you do that than delaying the debt you're paying off right now.