Grace saved $500 at the end of every month in an RRSP for fiveyears and thereafter $600 at the end of every month for the nextthree years. If the investment was gro...

The amount of money, $A(t),$ in a savings account that pays $6 \%$ interest, compounded quarterly for $t$ years, with an initial investment of $P$ dollars, is given by $$ A(t)=P\left(1+\frac{0.06}{4}\right)^{4 t} $$
If $\$ 800$ is invested at $6 \%,$ compounded quarterly, how much will the investment be worth after 3 yr?

So we're given the formula for amount after 30 years. Component Quarterly at 6% interest yes, A F p is equal to p times one plus 0.6 by four to the power 40. He's amount invested. He is the time in years. So we're giving piece $500 and he is two years. So monitors okay to the power for him to do it is approximately 5 63 $0.25.

It's a two parter, uh, for the first thing we're doing. Annuities deal. Um, because he's deposit at the beginning of each period, she's depositing an amount of $2435. We don't know the final value. Yeah, and this first slide onto the final value for eight years because that's all she's making these deposits. Uh, the interest rate is 6% but because its compass semi annually, we're gonna make it 0.3 for a calculation and the number of periods that semi annuals we're gonna do two times the number of years. It'll be 16. You're in value, can out for annuities due. Our final value is equal to our times Are interest rate 111.3 to the end plus one power. So 17th power minus one, divided by our interest, Rain Joe 0.3 And then we're going to subtract the our value from it to strike the final payment. Um, and so that that payment weakness to do a minus one inside are parentheses, so that it distributes out to the are Okay. So putting in 24. 35 for our I can see my final value. It's plug it into the character right now. Said clean for 35 23 £16. 17 power. That about 0.3 minus one. Okay, so the summit for eight years will be 50,000 5 54 on 40 and seven cents. And I believe that decibel in there because I believe the second part of the question, uh, she leaves that money alone so she doesn't take it out and re deposited, so she leaves it in there. So still at that decimal? Uh, she's gonna leave it in there for another five years. So now we're going to a compound interest to see where we're at after the next five years. Can't Teoh our compound interest. We have amount equals that principle, which is the amount in the first line times one plus our interest rate divided by end. We already have that sets 1.3 in the number of periods for five years. This is gonna be 10. Okay, so we're gonna take that I'm out and multiply by 1.3 thanks to the 10th power to the final amount. Shall having this account playing in the same number. From the first slide to this equation, it's gonna be 67,940 and 98 cents, maybe 97 cents at the bank. Forces around, Down. Okay, thank you very much.

So we're going to use this formula and it is not the perp formula. Uh I don't know if it's obvious to you, but the key word that I look for is the phrase compounded monthly Because then that tells me that n equals 12. There's 12 months in a year To use this formula. Not the perfect formula. The P value is the 6500. They gave you. The r. Value Is .065 for 6.5%. You have to move that decimal over twice. And then T. And the problem is three years. So we're ready to just jump right into this problem. And the future values A 6500 one plus .065 divided by 12. The end value. and uh maybe just do 10 times T in your head. Otherwise you might have to use parentheses when you go to your calculator. No matter what though you do need, you'll probably need to use your calculator because I don't know very many students that it would be good at .065 divided by 12. We're taking that to the 36 power. But when you use your calculator you should get the correct answer of $7,895 and 37 cents. If you round it the way I would round it to two decimals. But if you have a teacher Um that prefers three decim would be .366. But I'm going to leave the answer I like right here. No different answer here. What am I looking at? Problem 40. I have the wrong interest rate, don't I? Oh rats. Alright let me fix this real quick scratch that this is uh scratch that .0625. So I got to change this .0625. So when I go to my calculator as have to insert second elite A two In there, Ariel. Now I'm getting an answer that matches seven, or 68 to a few rounded three decimals. But there we go, so Nate threes.

Okay. This question wants us to determine the amount of money Rachel will have in her retirement fund under these conditions. So the first thing we should notice is that our A p R. Is divided into interest. That's compound in monthly. So we should probably convert everything in two months. So assuming she pays equally every month, 1500 a year, we're going to divide that by 12 to get 1 25 a month. And then if she opens it at age 19 and we care about home, which she has at age 55 then that's a 36 year gap. And 36 months. 36 years translates to just multiplying it by 12 432 months. So now we see that everything matches. So this is in months. This isn't months, and this is in months. So now we can work on finding the value. So were adding a sequence in the form P Times one plus the a p r. Divided by 12. Because we're calm, pounding it monthly race to the end. And in our case, we're paying $125 every month, and the A. P. R. Is 8.2%. So, in our case, the value of the retirement fund, we'll just be the some from the start, all the way through the 432nd month of interest of 1 25 times our common difference, which is 1.0 6833 Race to the end, and now we'll just evaluate this like a geometric series. So we use our formula where we put the first term times one minus our ratio raised to the number of collections divided by one minus the ratio. So in our case, the value of the retirement fund will be $328,000 374 and 10 cents. So that's a very good amount. So now it wants how much is interests? Well, the interest is just the value, minus the number of payments times the amount paid. So it's just our total value from the annuity minus 432 payments of $125. So the amount of it that came from interest was 274,000 374 dollars and one cent for 10 cents, rather so as you can see. Most of this fund came from interest and all right, this a bit neater, but that just goes to show the power of compound interest, and now we're done.