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MATH 1332/0132 521 eworkld -5812522208questionld-132l05k8 JlnttoAel lomework: 8.1 & 8.3 [graded homework] icore: 0 of 29 0f 33 (27 complele) '.3.23 Determi...

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MATH 1332/0132 521 eworkld -5812522208questionld-132l05k8 JlnttoAel lomework: 8.1 & 8.3 [graded homework] icore: 0 of 29 0f 33 (27 complele) '.3.23 Determine the present value YOU must irlvest: havc {ulute value A al siple interesi rate atter linie ' A = $11,000,r = 3 0%,t = eals The present value Ihat must be invested get 511,0D0 after ycars an imeres rate of 3 0%8 /s (Round upfo Ife nearst Cert )

MATH 1332/0132 521 eworkld -5812522208questionld-132l05k8 JlnttoAel lomework: 8.1 & 8.3 [graded homework] icore: 0 of 29 0f 33 (27 complele) '.3.23 Determine the present value YOU must irlvest: havc {ulute value A al siple interesi rate atter linie ' A = $11,000,r = 3 0%,t = eals The present value Ihat must be invested get 511,0D0 after ycars an imeres rate of 3 0%8 /s (Round upfo Ife nearst Cert )



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Interest on an Investment A 3 -year certificate of deposit pays interest at a rate $r$ compounded annually. If $\$ 2000$ is invested, then the amount at maturity is $$A=2000(1+r)^{3}$$ a. Expand the expression for $A$. What is the degree of the resulting polynomial? b. Find the amounts $A$ for the values of $r$ in the table. $$\begin{array}{|l|c|c|c|c|c|} \hline \begin{array}{l} \text { Interest } \\ \text { rate } r \end{array} & 2 \% & 3 \% & 4.5 \% & 6 \% & 10 \% \\ \hline \text { Amount } A & & & & & \\ \hline \end{array}$$

Yeah. In this question we are dealing with a certificate of deposit that is being offered by just bank. So certificate of deposit is an investment instrument where you have to keep the investment for a certain amount of period. So here the thing is it is it is a 36 months, 30 minutes. So what that means is that you have to keep the money mr for third at least 36 months only afterwards you can withdraw the amount. Okay. So the interest rate offers 2.25 So uh solution this is a component of this problem. So the question that we have to use this effort is going to be multiplied by one plus R. Buy into the bottle empty. That is what we need to use. Well it is the time period for which the investment is kept in the account and there's a number of components done nine year R is the rate of interest. That is to be extras used in the smell is not in person is pc principal investment and a is the amount that you get at the end of this particular time. 20 years. So usually it is seven years but if it's not then it will be given a specified, okay you're sleeping second year. So our is given personally. So you have to convert into the small value. So audio clues that you started is divided by 100. So that was Robin 0 to 25 and we don't have the value of them. So what we can do here is that we can assume and equal to one. The reason is that companies generally is what we can assume because First of all it's not specified in the question that company has done 12 times a year for four times a year. So and even if it was uh we won't be able to see a huge difference in most of the problems. So assuming any gold one as it's not given. Makes the whole calculations simpler. And yeah that's the main point. So We can go with annual one. So the first question, the first question if physical 2000 physical to 2000 please. 2000 orders A. Of eight. That means if you're hiring mr $2000 at the beginning at the start then what will be the amount that will get at the end of eight years? That's what we need final. So that means t is equal to eight right? So you have to lose and use how to substitute these values in this equation. And finally A fate. So that means a. of 80s. It will do 2000 multiplied by one plus 10.2 to 5 divided by one. To the borrow one multiplied by eight. So during the calculation we gave 2000 multiplied by 1.2258. uh there will be 2300 and $23.89.66. So this month will be the amount that you will get at the end of eight years. Okay so from this 30 year deposit now moving to the second question that is be solely going for a F. D. Is equal to 4000 a of things that will do 4000 and we need to solve it 40. So what does t. So here's the thing is if you take the situation you need the value of P also to solve this. Right? So here we don't have the value of that has been principal investment. We don't help. So what we can do is that we can just assume that the principal investment is 2000 and go forward with with it just He is equal to 2000 is something that began, Take it from the previous question and then it's all about 40. Another method but you can use that lou PSP and then solve this equation and you will get an equation. 40 in terms of P. Okay so either way you can do that I'll do both. So first I'll go with the 2007 August 2000. So oh that is 4000 is equal to 4000 is equal to 2000 Multiplied by one plus .02-5, divided by one to the bottom one multiplied bhakti right, dividing both sides by 2000 will get 1.0 to 251.2 to five to the power of T. Is equal to to write 6000 and two. And playing the algorithm you're gonna play. Nice. Look at the work Common law with uh so you want no problems. I'll use common law with log 1.02-5 to the side of things. It will do long too here you can use the property that is uh vlog next to the bar of physical too. A log X. You can use that. So that would be T log 1.0 to 250 will do Log to write, dividing both sides by log 1.02-5. You get these, you can do Long two divided by log 1.25. And this will be Long to Development Long 1.0- 1231.31 .15. That means it will take around approximately 31 years for the amount to grow from $2,000 to 4000. So in 31 years they're going to be doubled in this case that is 1 2000 and others. Now if you have to do according to the you know without a single Afobe the questions 4000 is equal to p multiplied by one plus people played by Mhm. He multiplied by 1.2 to 5. To the bottom of the right here will have to be here. Now play along with um on both sides. Just like what we did here by log open. So log 4000 is equal to here. We can use the property that is uh yeah log M. N. Is equal to log tempus. No you can use that and that will leave you log T. Plus log 1.2 to fight Gaddafi. But this can be written us T log 1.0 to fight due to this property. Right? So log P plus T log 1.0. Right? Yeah. Mhm. T Log on 10- 2 5. And here we need t to be isolated because he's the value that we need to find. Right? So we'll take it with a backlog. Be on both sides and divide both sides. Very long. One point on the road to fight. So that means T. Is equal to log 4000 miners log B. Only one of my log 1.2 to 5. And here another property that union uses that log m minus log in log and minus log in and log and brian. Okay so once you use that thing he will do log 4000 divided by B. Only one of mine Law 1.0 to be quite. Okay So this is the equation that this this can be an equation. Well you can get it if you know the so if you pay us 2000 then you will get t. So the thing is here the amount at the end of I'm not at the end of two years is two years is $4,000. Okay so This is applicable only for $4,000. So if you want to change the equation then you will have to change the amount of the amount that is expected or do the necessary changes. Okay so now I'm going to be third question. Third question that is what principal amount should be invested to that account balances $2000.03 years. So there are a lot of trees. We will do $2000. If this is the case then these you will do what that's what we need to find out. They are given that is we will treat right? So you can do this by you know this equation. This equation where you have you can You have to find T. Which is straightforward. All you can use an equation like this which is also another form of arboreal equation. But the thing is you have the instead of 4000 you will have to use 2000. Okay so that's what you need to do. So if you want you can buy that also a problem and I will just go through this way. So that means 2000 is equal to F. D. That is 2000 and two P. Multiplied by one plus 0.2 to five divided by one. To borrow three multiplied by one. Right? That is the question after substitution. So that means 1.023 1.0 1.02-5. That is this 1.02 to fight to. The part of three multiplied by P. is equal to 2000. So that means to be a. Is equal to 2000 divided by 1.02 to fight developed. And that will be 2000. There one x 1.02- five. To the power of three. Yes. Around 101,870 A. Mhm. $686. $85. Mhm. $1,870.85 is the principal amount that you should have invested. If you want to get $2,000 at the end of three years in this 30 ft of deposit. Now to the final part that is we need to find the simple interest rate that uh it turns the same amount of interest after one year as the cardboard injustice. So you know that simple. And just the in the case of simple interest, if you have a principal amount then the interest is calculated on this principle amount only. Okay so but in the case of compound interest, this principle investment will get increased. I mean if you're for example if you're just investing $100 in an account, $100 in an account. Well you have two options. The first one is common interest. First one is okay, okay simply dress and the second one is home board interest. Okay sunday. Uh First year. If it gives you uh if this gives you An interest of 5% is Parana 5%. So that means if you invest $100 in the first year you will get an interest of $5 right in both accounts In the 2nd year. Even though your account will have 11 or 51 or $5 105 dollars In the case of simple interest you will only get you will only get $100. The anticipation, $100. Because that is the initial investment. Okay so interested again you'll get $5 but there is a compound interest. The interest is calculated. Don't be total balance in your compartments. The initial investment and the interest you earned. So the interest is calculated on one or five. So when there's no one not five it will be 5.25 will be given to you. Okay $5.25 will be given to. So when it comes to 30th You even though you have, you will have 11115. 110. 110.25. Even though you have 110.2 when your balance your account you'll be given $5 in the case of simple interest. But here the interest will be in different rate and that you need. Uh Yeah 5.5 and 255.5 and 25 the special B. D. Interest that will be earning in that year. So that means compound interest works on the total capital at your account help. But in the case of simulators, the industry's capital around the initial investment. So what they're asking is that in this particular case, what would be the annual person is healed or the world being a simple interest? What will be the equal in simple interest for this particular account? That's what they asked. So we can just say that for this party like when we have a equal in simple interests so that you will get this much but but we can only say it for a particular time period. For example If the investment is for 36 months We can say that for 36 months. The equal in simple injustice exposing this Equal intent. Simple sentences expressing this is what we can say but for 48 months it won't be expression is it can be by personally so you have to calculate that for every time period. Okay so I'll hear do here an example. So usually what we have is that it's a 36 months or direct deposit. Right? So we'll take 36 months or three years And three years. We have 2.25% is as compound interest. So how much would be if you're investing if the principal amount is $2,000. The principal amount is $2,000. uh then a or three will be At the end of a of three will be 2000 Multiplied by 1.02 to fight to the bar off to the bottle. Uh Three right In the second will be three. And that will be 2000 multiple every 1.02-5 To the bottom of three. And you will have $2,102,138.06 in your account. But this is for three years. Okay. So the interest here is interest here is 2138 106. That is the final lemon minus the principal investment. 2000. Right? So this will be then 1 38.6. Right? So what this means is that even if the account uh even if the account is in simple and that you should earn 138.06 as interest at the end of 333 at the end of three years. For that? What you can do is that you don't tell the equal in some places where you can do is that you have to uh You have to divide this that that is 1 38.6 is the interest that you have earned in three years. Right? In three years. So that would be 1 38.6 developed by three will be 46.0-46.0. So you have to earn $46.02 as the interest every year on the capital that you haven't missed it. So that means this is the interest that you have earned On the capital of 2000. If it was simple interest. So how much person? It will be 4-6.202 developed by 2000 Multiply 100. Right? But maybe person is straight The right way to 2000. Only 500. That is 2.3012 point 301 person is. So what this means is that yeah four three years. Both for three years. For three years simple interest equivalent Is simple interest equal in this 2.3's Roman person is but for four years it can be different but for five years can be different. Can be it won't be 2.30. There'll be a slight difference. So for that you have to do the calculations or or the one thing that you can do is that one thing you can do is that you can compare the equations of both. Simple interest and compound interest. For compound interest we have a ftse will do p multiplied by one plus r by N. To be bought off in Tikrit. And for simple interests we have the equation. A. F. T. Is equal to be multiplied one plus R. P. Right be multiplied by one plus happy. This is the equation for simply because where there is the amount that you get at the end of two years. These the years under investment and then we will be the principal amount. So what we're assuming here is that at the end of this particular time period you will get there same amount. So that means A F. D. In both cases are the same. So left hand side, Alexis the same. The four equipped artist. So when nuclear artists be multiplied by one plus are by and to the core of Mt. So let this be our and let this be up. Are we okay? Is equal to hey multiplied by one plus or one. So here T. Will be this thing because you're investing for the same time. Great then uh principal amount will also be the same. So as we have principal I want same on both sides. We can we can you know divide both sides by B. And will yet mhm. One plus our brian are buy in to the power of M. T. Is equal to one plus R. One T. So that is the case. You know the value of our right ours to point I was 2.25. So subscribing the value of R. And then you will get 1.2 to 5 to the power of T. Is equal to one plus R. One T. Okay so you you need to know the value of our one. So that means you have to subtract one from both sides and divide the whole also so that means our one is equal to 1.02 to fight to be part of T. Once wrote to go T -1 whole developed by T. So this will be the interest rate that you the quick and simple interest rate that you will get uh from the support account. So using this question one calculate that with the time period that will be you have to give you the time period that is three years or four years or five years and you'll get the equality interest rate. So this question is for the second and for another accountable how to form the question by following the same steps that is equating these two equations and then arriving on this Relation. Okay so when seen through the steps. So here we have an account with a positive uh this is completely interested uh with the rate of 25% family. So we assume that the components and annually. And the first question we were asked to find the Returns at the end of uh eight years. So we found that and in the second one we're asked to find me Uh sold the question 40. And we didn't do waste in the first one we found the amount of 50 years it'll take for you know for $2,000 has been able to become $4,000 It took we found that will take 31 years. And the second one we found the equation that lets TMP were please not given right. And the third one we were asked to find the principal amount That is if you won't be investment in three years to go to $2,000. So we found that you have to invest $1870.85. And the fourth one we were asked to find the relation between this simple interest and the component status. We need to find the simple interest equal of this world will account. And we found it using two ways. So for 36 months for three years, if 2.2% was the component list then we found that by calculating the interest earned and dividing it by three And finding the percentage of that party I want to implement. We found that the current for three years is 2.301 person is but if you want to have a general relation between these two then you have to equate the equations of uh compound interest and simple interest. And after doing that we found the list that are one with our one is easy Are one you see simple interest equal and anti easy time period. Okay so that's it for this question. I hope it scared like

So let's say that you would like to get $75 after three years at an annual interest rate of 8%. Compound ID. Quarterly. How much would you need to invest right now in order to get $75 in three years? So since this is being compound ID, we would use the compound interest formula or a equals P that was one over R Oh, and to the end, take power where a is the future value. So in this case, it would be $75. He is a present value. That's what we need to solve for our is a great or 8% and as amount of times that's being calm, pounded. And since it's being compounded quarterly and would be four and the time is three years, okay, now that we have all our values, we can plug them in. So we have. But so we have 75 for a equals p times one because 0.0, it over, boy to the four tires, three power. So we can divide 75 by all this. So we have 75 over one plus zil point Go aid never before to the 12th Power equals P. I know this looks pretty awful, but all you have to do is plug it into your calculated Just remember, um, proper apprentices so that everything is divided correctly. One plus, so we get a p equals $59. Oh, and 14 cents. So that means that you would have to invest $59.14 right now to get your $75 in the future.

So we can use this formula for interest equals P R T were He is principal. Ours rate and t is time. Um, so we had $221 upside handwriting there. $221 an interest. Um, we know the principal was 3400 and we're trying to figure out the rate if that interest was accumulated in just one year. So basically, we just need to divide both sides by 3400 and we'll have the rate. All right. So to 21. Divided by 3400 his point, or 0.65 that is 6.5%.

We know that Margaret invested a certain amount of money, Let's say X dollars at 4%. And she invested $3000 more than that. So x plus 300 I mean 3000 at 6% and that all total in interest. She got $780. So 0.4 x close 0.6 x plus 180 equals 780. So add those. Together we have 0.1 x. We can also subtract the 1 80 equals 600. If we divide by 1/10 the same is multiplying. By 10 we'll get 6000. So we know she invested $6000 at 4% and 3000 more than that would be $9000 at 6%.


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