) Helen is thinking of buying a new car: She has to pay 53800.00 down payment and S500 at the end of each month for the next 8 years Calculate the cash value of the...

Buying a Car Amanda Perdaris wants to have a $\$ 20,000$ down payment when she buys a new car in 6 years. How much money must she deposit at the end of each quarter in an account paying 3.2$\%$ compounded quarterly so that she will have the down payment she desires?

Okay, so you have that Christie puts in. So that's RP 10,500 into account that she wished she wishes to. I'm saved for 12 years, hoping that car would cost 30,000 by then. So that's a is you got to do that and ratifying our great. So that's arm is equal to question Mark given, that's a We have compounded quarterly. Which means that every single to for our equation A is equal to p. Times one plus R over m to the power, uh, 12 times for and then for B, we have compounded continuously. So that's a is equal to P, which is 10,500 times each and five or and T, which is 12. Okay, so let's solve for R both of these cases. So we have also a of 30,000 that I forgot to put it. So let's put that end. So am 30,020 and I would solve for R. So we're gonna divide by 10,500 on both sides. Now we have the following to the power of 12 cents four. That's, um, 48 now on multiply both sides by 1/40 eights to isolate our okay, so I get dirty. Death in divided by $10.500 didn't have won over 48 is equal to one plus R over four, and now all subjects one on both sides and then multiply by four and we get the following. So this minus one and multiply what both sides before. So we get four minus four here is equal to our and I would solve for R to get four times 30,000 divided by 10 500 to the power of 1/48 minus four. Which gives me a rates off 0.88 And that's equal to eight points eight cents for art rates. And what about if the compound continuously so dividing by 10 500 on both sides that you could eat that part on Times 12 and I will take the eligible science to get rid of our base of E. That's even to our times, calls the writing by 12 on both sides, we get on one of 30,000 divided by 10 500 then divided by 12 which gives me rates off 0.87 which is pervades of 8.7%. It's

41. I have a wolf. Are you hurt by one minus one? Plus I in the end with negative and power where a equals 18,000 r equals 3 75 and n equals five years Fortune 16 months. So now I'm plugging in my known quantities I have 18,000 equals 3 75 eyes with my ex one minus one plus x with negative way Multiply both sides by the Super Bowl of 3 35 over X. That would give me 48. Yes, equals one minus one over one plus x the city of power multiplying everything by one plus x to the 60 of power I get 48 It's one plus X with the 60th power moves one plus x to this power Buying this one. Bring everything to one side and I have boarding in X finds one plus x to the sixties Power mine It's one plus X with 60 of power plus one equal zero, which is my ex. And now I'm gonna take the derivative of that. No, my prime. I take the product mobile. I'm gonna get 48 finds one plus to the six of negative 60 of power. No positives. You have power Loves 48 x time 60 one plus X to the 59th Power minus six with one less eggs to the 59 hours. So my f fine. It's been a simple morning. A one plus X in the 60th. No. 28 baby one plus X with 59 power minus 60 one plus X to the 59 power. Now I'm gonna factor out right with the factor out at 12. One plus eggs to the 59th Power You can't leave me where Work one plus x plus 2 40 2 40 Thanks, minus five. Distributing that four. So I get a crime equals O on X with 59 hour combined My life terms Wait 2 40 or were X minus one. That is my f frying. Now I get to Newton's equals x So start with 1%. So I get X equals 0.1 except two Quite 008 due to go to except three equals, You know, row seven six, you know to exit four 76 two and exit five equals at six, which so six 286 is 0.76 286%. And that would be our monthly. Never want to figure that out. Compounded her year you were just multiply that fight both. No compounded monthly per year. Compounded monthly.

Okay, so this problem wants us to find the future value of an ordinary annuity case. Of the payments being made are R is equal to $1250. Our interest rate is 5% and the number of years is 18. The only difference with this one is it's ah, semiannual, Ah, com pounding s. So that means we needed to multiplier in by two. Since it's happening twice per year, so in for the formula will be 36 and then our high, because it's having twice a year, is gonna be divided by two. So be 2.5%. So our formula for the future value is s equals R R 12 50 times one plus our interest rates of 1.25 not to the ants powers to the 36 power minus one divided by our interest rate, which is 0.25 Okay, it's gonna be 12. 50 and I'll do the decimal part and a calculator. So 1.25 to the 36 power a minus one and then divided by 0.25 because that body is gonna be 57.3, and I'm gonna keep that number in my calculator. Just go is a repeating decimal and remote by about 12. 50. So after multiplying by 12. 50 the future annuity amount be at 71,000. Uh, sorry. 71,006. 26 0.7 seven. Okay. And so that's the amount that will be made on this. Uh, the other part of this problem wants us to find out how much how much will be made. Uh, just are deposits are contributions. So to find out what is going to 12 15 times, 36 contributions. Okay, so 12 50 times 36 people to $45,000. Okay. And then So that's how much is is by contribution. So that's one part of our answer. Our total amount was this 71,000. So to find how much is made by interest, we just need to subtract those two numbers. So you have the 71,000. We're gonna subtract 45,000 the amount made My interest is 26. Uh, 626 0.77 Is our interest payment? Not bad. Okay. Thank you very much.

Okay, this is a compound interest problem. And they have a kind of simplified interest formula A equals P one plus I. To the end, i is not the annual rate. It's the annual rate divided by the number of times compounded per year. And then in is the number of times compounded. So they asked if Dave deposits $10,000 at an interest rate of 6% compounded semi annually. That means I get 3% each time it's compounded. How long will it take to make to make it to $11,000? So I divide both sides of this equation by 10,000 and I get 11 10th is one point oh three to the end, take the log of both sides and I get log base one point oh three of 11 10th is in, so in is 3.22 times compounded. Have to compound it three times and I won't have enough. So I need to compound it actually four times. I only compounded semi annually. That means twice a year. So my answer can't be 3.22 I have to round it up to the next time. This is the number of times compounded but I compounded twice a year. So if I compounded four times that's two years to earn 11,000, let her be says that after a certain amount of time the amount owed is $1241.18. The amount I started with was $1200. And the interest being charged on my over to overdue balance is 28% compounded daily. And the question is, how many days was it compounded to reach that 0.1241 point 18. So I followed the same processes before divide by 1200 12 41 18, divided by 1200 equals one plus 28 over 3 65. Mhm 0.28 is 28% divided by 3 65. And all of that to the end. So my answer in is gonna be log of 12 41 18 over 1200 divided by log of one plus point 28/3 65. In comes out to be 40.4 96. But I can't compound it 40.5 times so I need to compound it 41 times. So the number of days in this case the times are days. The number of days it was compounded is 41 days. Okay Part C says how long does it take at 5.5% compounded semi annually or an investment to triple Yeah but they didn't give me the initial amount but they don't have to let the initial amount Bp and then triple that amount. The final amount has tripled the original. So the original is p. The final amount is three p times one plus point oh 55 divided by two raised to the in this last one wasn't 40.496 This one should been 44 days, copy that number wrong. That should be 44 days. When I divide by P. I get three equals log, sorry, I get three equals one point oh 275 to the end. Take the log of both sides and I get log of three divided by log one point oh 2 75. That gives me 40.4 96. I'm compounding it semi annually. I can't compound it 40 times 40.5 times. So I compound it 41 times. That's 20.5 years invested before I triple my money. Thanks for listening to have a great day.