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so let's generalize a bit what we learned in the last presentation let's say I'm borrowing P dollars P dollars that's what I borrow so that's my initial principle so that's principle principle and let the R is equal to the rate the interest rate that I'm borrowing at or we could also write that as a hundred our percent all right and I'm going to borrow it for well I don't know T years T years let's see if we can come up with equations to figure out how much I'm going to owe at the end of two years using either simple or compound interest so let's do simple first because that's simpler so at time zero so let's make this the time axis time zero how much am I going to well that's right when I borrow it so if I borrow paid it back immediately I would just dopey right at time one IOP plus the interest plus you can kind of view it as rent on that money and that's R times P and that previously in the previous example in the previous video r was ten percent p was 100 so I had to pay ten dollars to borrow that money for a year and had to pay back one hundred and ten dollars and we can this is the same thing as P times 1 plus R right because you could do this as one p plus RP and then after two years how much do we owe well every year we just pay another RP right in the previous example is another ten dollars so if you know if this is ten percent every year we just pay 10 percent of our original principal so in year two we owe P plus RP that's what we owed in year one and then another RP so that equals P plus 1 plus 2 R and just take the P out and you get a 1 plus R plus R so 1 plus 2 R and then in year 3 we'd owe what we owed in year 2 so P plus RP plus RP and then we just pay another RP or another say you know R is 10 percent or 50 percent of our original principal plus R P and so that equals P times 1 plus 3 R so after T years after t years how much do we owe what's our original principal times 1 plus and it'll be T R so you can distribute this out because every year we pay PR and there's going to be t years and so that's why it makes sense so if I were to say I'm borrowing let's do some numbers I find you could you could work it out this way and I recommend you do you shouldn't just memorize formulas if I were to borrow $50 at 15 percent simple interest for 15 for let's say for 20 years at the end of the 20 years I would owe $50 times 1 plus the time 20 times 0.15 right and that's equal to $50 times 1 plus was 20 times 0.15 that's 3 right right so it's 50 times 4 which is equal to $200 to bar it for 20 years so $50 at 15% for 20 years results in $200 payment at the end so this was simple interest and this was the formula for it let's see if we could do the same thing with with compound interest let me let me erase all of this that's not how I wanted to erase it there we go okay so with compound interest in year one it's the same thing really as simple interest and we saw that in the previous video io P plus and now the rate times P and that equals P times 1 plus R fair enough now this is year 2 is where compound and simple interest diverge and simple interest we would just pay another RP and becomes 1 plus 2r in compound interest this becomes the new principal right so if this is a new principal we are going to pay 1 plus R times this right our original principal was P after one year we paid 1 plus R times the original principal write times 1 plus R rate so to go to into year 2 we're going to pay what we owed at the end of year 1 which is P times 1 plus R and then we're going to grow that by our percent so we're going to multiply that again times 1 plus R times 1 plus R and so that equals P times 1 plus R squared so the way you could think about it in simple interest every year we added APR like you know what we ever in simple interest we added plus P R every year so if this was you know $50 and this is 15% every year we're adding $3 we're adding was at 50% we're adding 750 and interest right where P is the principal RS or 8 in compound interest every year we're multiplying the principal times 1 plus the rate right so if we go to year 3 we're going to multiply this times 1 plus R so year 3 is P times 1 plus R to the third so year T is going to be principal times 1 plus R to the teeth power and so let's see that same example this is the Rio $200 in this example with simple interest let's see what we owe in compound interest the principal is $50 $50 1 plus and what's the rate point 1 five and we're borrowing it for 20 years so this is equal to 50 times 1.15 to the 20th power the 20th power now you can't read that but let me now let me see what I can do about the 20th power so let me use my Excel let me clear all of this because I just know actually I need to just use my mouse instead of the pen tool let me clear everything okay so let me just pick a random point so I just want to plus 1.15 to the 20th power and you could use with any calculator sixteen point three seven let's say so this equals fifty times sixteen point three seven and what's 50 times that plus 50 times that eight hundred and eighteen dollars eight hundred and eighteen dollars so you now realize that if someone's giving you a loan and they say oh yeah I'll lend you you need a 20 year loan I'm gonna lend to at fifteen percent it's pretty important to clarify whether they're going to charge you 15 percent interest at simple interest or compound interest because with compound interest you're going to end up paying I mean look at this just to borrow $50 you're going to be paying six hundred and eighteen dollars more than if this was simple interest unfortunately in in the real world most of it is compound interest and not only is it compounding but they don't even just compound it every year and they don't even just compound it every six months they actually compounded continuously and so you should watch the next several videos and continuously compounding video interest and then and then you'll actually start to learn about the magic of e anyway I'll see y'all in the next video