Suppose Peter opened a savings account on January 1, 2000, and made a deposit of $5100. In 2001, Peter began depositing $51 into the account at the end of each mon...

If a savings fund pays interest at a rate of $6 \%$ per year compounded semiannully, how much money invested now will amount to $\$ 5000$ after 1 year?

We have a body, you know, we have to find his six. This is even as 500 into one minus when you needed one for work. 1000 to the power of six. Fuller One minus wanted you dead. You'd one of what? 1000. So after solving reopen, this has treated Raju 7.51 Dogus. So this is the amount. Which is what he did off the sixth it wanted. And the amount it was dead before six. The budget is equal to Donna trees. It was little. 7.51 minus Gotta 500 is equal to Darla. 2507.51 causes dancer for part A off the wrong in body with refined is well, so Ace two equals 500 into one minus one. Geological one over 1000 to the power off to L. A. Well, one minus. I want you to do one over 1000. So after something, we often this and Stahler 62 to 3.112 This is I have to. It was doing You have to want it and we have to complete the amount. It was dead before Dr Deposit because It's equal time. Six years True. 3.112 minus dollar fine 100 pointed You know, this is equal to God. Fine, Fine. 23 0.11 is a dancer to be given problem.

Alright, so here I have kind of the setup for this problem, we have our information in black, they gave us 3% per year that we're going to grow in this fund and it's compounded semi annually. We wanted to result in $5,000 after about a year. So we're going to use the compounding interest formula here. A is the final amount. P is the starting amount times one plus R. Is the rate divided by the number of times compounded raised to the number of times compound at times T. So n happens twice in this formula 3% rate. That's my are. But I'm going to use .03 semi annually means I'm cutting this in half. Like I'm compounding it twice a year. So I'm gonna use to for compound for the number of times is compounded. The goal is to get to 5000. So that's the a. And the time is one year. So I can set up this equation right here with all that information. and then at this point I can put everything on the calculator to be able to solve some. Just put one plus Point of three divided by two. And the order of operations will take care of that for me Race to the two times 1. So I get that decimal 1.3 oh 2-5. And I want to keep as many of those numbers as possible. I'm going to divide by that long decimal and that's going to get me my final answer that I need to accomplish. So this number is how much my original mounts can get, multiplied by over the course of the year To my back and press two with this negative button to get answer. And it's gonna copy that decimal in for me. So 8 4053 and 31 cents. And that is my answer.

Okay. So here I'm gonna be using the compound interest formula where we have A. Is equal to the principal. P times one plus are over mm raised to the end. So here we have that P. Is 4000 R. Is 6%. So that 0.6 and M is 52. Um And N. Is one half time 52. So N. Is 26. So therefore part A. We have that the amount A is going to be equal to 4000 times one plus 0.0 6/52. And then we raise that to the 26. And we get here that this is going to be approximately equal to $4121 And 75 cents. So therefore the amount in the account here is gonna be $4121.75. And then four parts b well now um we have again ps 4000 hours 0.6 52. But now N. Is 10 times 52. So N. Is going to be 520. So now we have 4000 it's fourth 1000 uh times one plus 0.06. Mhm. My computer is kind of freezing here. Alright play 1.006 over 52. And then we raise that now to the 520th power. Yeah. Mhm. Mhm. Mhm. All right, says um I think Might be in the sun here and having a little bit issues with my computer. Okay. We raised this to the 520th power. And then we get the amount is going to be 7285. Uh $7,285. So our amount in the account is going to is going to be equal to so I can get it here. It's going to be equal to. Oh, Alright. I apologize. I'm having some issues um with my computer right here. But again, the amount is $7,285.95.

Okay in discussion, some one. Diego deposited a certain sum of money in bank two years ago. So the time period till now it is two years. Okay. And if the bank had been paying interest at the rate of 6%, so our is six per annum. Okay. And it is compounded continuously. No problem. And he has total $12,000 deposit today. So we can say the total accumulated amount is $12,000. Okay. And we have to find out what was his initial deposit. So we have to find out the principal amount that he had deposited two years ago. Okay. So we know it is compounded continuously and the formula of uh formula of compound continuously, that is accumulated value because two P that is his principal amount here is to the power artie. So from here is principal amount. It will be a multiplied by it raised to the power minus RT. So we can say P is here, it will be 12,000. Okay. He raised to the power minus RT. So negative are a 6%. That is 0.6 and t is time period. That is too. So it will be P equals to 12,000 multiplied by it raised to the power negative 0.12 Okay. And when we saw it was it is $10,643.05. Okay. It means the percent they go and deposited Diego deposited $10,643.05 2 years before today. So that today it's total deposited Total deposit is $12,000. Thank you.