For each of the scenarios given in Exercises 1-6,

Key Concepts

Doubling Time

Doubling time is the period it takes for an investment to grow to twice its original amount under exponential growth. It is typically calculated by setting the future value equal to twice the initial investment and solving for time using logarithms. Understanding doubling time helps in quickly estimating the impact of a given interest rate on investment growth over time.

Average Rate of Change

The average rate of change measures how the amount in an account changes, on average, over a specific time interval. It is computed as the difference in the account balance between two points in time divided by the length of the time interval. This concept is used to assess the performance of an investment over particular periods, such as short-term versus long-term changes, providing insights into the investment's behavior over time.

Compound Interest

Compound interest is the process where interest is earned not only on the initial principal but also on the accumulated interest from previous periods. It is calculated using the formula A = P(1 + r/n)^(n*t), where P is the principal, r is the annual interest rate expressed as a decimal, n is the number of compounding periods per year, and t is the time in years. This concept is crucial for understanding how investments grow over time when interest is compounded at regular intervals.

Exponential Growth

Exponential growth refers to an increase in a quantity where the rate of growth is proportional to the current amount. In financial contexts, investments increase exponentially when interest is compounded, meaning that with each period the amount increases by a factor greater than one. This growth is represented by an exponential function and is fundamental in predicting future values of investments.

Transcript

00:01 In this question we're talking about compounding interest where interest is being given on the interest that is already given for the principal amount.

00:13 And in this question, compounding is done every month, okay, for a principal amount of, for a principal amount of $5 ,000 at a rate of, at a rate of 2 .125 percentage.

00:29 This 2 .125 percentage now we told that the compounding is done continuously that means number of compounding in a period is in a in a nearest 12 right if it is being done every month there will be 12 times it will learn and the equation that we need to use is a of t is equal to p multiplied by 1 to the 1 plus r by n whole to the power of n t here t is the number of years for which the particular investment is kept in the account and p is the principal amount that is being invested in that account r is the rate of interest which has to be expressed in terms of design numbers not not in terms of percentage and n is the number of compoundings done in an year well a of t is the amount that you can expect for the account to hold at the end of t years okay so the principal amount are and n but here we should not take in percentage that means we have to divide it by 100 so that means 0 .02125 is the amount that you take is the value that you need to use so the first question is to to express the amount in this particular account in terms of t so all you have to do is that you just have to substitute these values into this formula and write the equation so that means a of t is equal to p that is 5 ,000 multiplied by 1 plus rs 0 .02125 divided by 12 1 to the power of 12 t okay so this is the formula that represents this particular account now coming to the second question that is how much will be the amount that is that will be present in this particular account at the end of at the end of 50th year 10th year 30th year and 358.

02:34 So all you have to do is that you have substitute t in this formula and find the answers.

02:40 So that means a of 5, the first one, that is on the 8th of 50 is 5 ,000 multiplied by 1 plus 0 .02125 divided by 12.

02:52 What do we borrow? 12 multiplied by 5.

02:56 So that means calculating that we'll get 5 ,000 multiplied by 1 plus 0 .02125.

03:04 Divided by 12 whole to the power of 12 multiplied by 5 right we'll do that we'll get 5, 5 ,559 .59 this much will be the amount that the account holds at the end of 50th year now at the end of 10th year multiplied by 1 plus 0 .02125 divided by 12 whole to the power of 12 multiplied by 10.

03:51 That means it will be 6 ,000, 6 ,182 .67.

04:02 This must be the amount the account will hold at the end of 10 years.

04:07 Now coming to 30 years, a of 30 is equal to 5 ,000.

04:14 Multiplied by 1 plus 0 .0215 divided by 12 to the power of 12 multiplied by 30 is equal to 9 ,9 ,9 ,453 .3 .3, yeah, we'll get 4.

04:38 Rounding it off, we'll get 4.

04:43 Okay, so that's it.

04:47 And that means air.

04:49 Of 40 if you want to be you know precise then 40 a of 35 is equal to 5 ,000 multiplied by 1 plus 0 .02125 divided by 12 that is to the power of 12 multiply by 35 and you will get 10 ,000 10 ,000, 500 and 12 .13.

05:26 One actually you will get one four yeah you get one four so just a minute you will get one for because you are asked to round of the answers to its nearest scent so that's how you will get one four no you will get one three anyway be careful about the rounding of rules so just have to make sure that it is correct okay so this will be the amount present in the account at the end of 50 will have 5 ,559 .98 and at the end of 10th they will have 6182 .67 at the end of 30th year we'll have 9 ,453 .4 and at the end of 35th year we'll have 10 ,512 .13.

06:34 That means why 35th year the money has already doubled.

06:40 That brings us to the third question that is we need to to find the number of years it will take for the amount for the amount to become double that is if the principal amount was the principal amount is 5 ,000 then the amount at the end of t years has to be 10 ,000 so you need to know how much it will take we know that by 35 years it has already doubled but we just need to do the calculation to know how much to take so substituting the values will get 10 ,000 substitute values to be to this equation okay you need to to buy this equation and you need to find t 10 ,000 is equal to 5 ,000 multiplied by 1 plus 0 .0 2125 divided by 12 by 12 to the power 12 t this is what we need to find t is what we need to find and dividing both sides by 5 ,000 we'll get 2 sorry 1 plus 0 .0 2 1 plus 0 .0 2 1 2 by 12 to the power of 12 t is equal to two now applying log rhythm you can apply a natural logarithm or common logarithm anything can be applied no problem so applying i'm using natural logarithm natural log of 1 plus 0 .02125 divided by 12 okay or we'll use common logarithm just for this case okay so it doesn't matter actually doesn't change anything much but i'm just you can use any base if you want if you have a calculator that supports different base then you can use for example you can use base to so that the right hand side becomes one no problem it's it's just that you can try using natural and common logarithm but when it comes to e you can you have you have to use natural logarithm right because it's simpler when it comes to 10 use common logarithm because it makes things simpler is equal to log 2 that means applying properties of logarithm we get 12t multiplied by log off inside the bracket will have 1 plus 0 .02125 divided by 12 that is 1 .1 .1 point we'll have a let's take the logarithm i think that would be better because if we just round off here we'll have it's it'll be a bit difficult log of that particular value is 7 .7 .68 multiplied by 10 to the bar of minus 4 is what we get is equal to log 2.

09:57 So that means 12 t is equal to t is equal to t is equal to log 2 divided by 2...

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