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

step 1 In this question we're talking about compounding interest. So, converting interest is given on t

Key Concepts

Average Rate of Change

The average rate of change measures the change in the value of an investment over a specific time interval, divided by the length of that interval. It provides insight into the incremental growth rate over a period, helping to understand how the investment's value evolves between two points in time.

Doubling Time

Doubling time is the period required for an investment to grow to twice its initial value. It is typically determined by solving the compound interest equation for the time variable, often using logarithms to isolate and calculate the time needed for the doubling of the principal.

Future Value Calculation

The future value of an investment is calculated using the compound interest formula, which accounts for the principal, interest rate per period, number of compounding periods, and time. This formula provides a way to predict the accumulated amount over a given time frame.

Interest Compounding Frequency

The compounding frequency determines how often the interest is calculated and added to the account balance within a specific time period. Higher compounding frequencies, such as monthly or daily, result in more frequent application of interest, affecting the overall growth of the investment.

Compound Interest

Compound interest refers to the process where the interest earned on an investment is reinvested to earn additional interest over time. It leads to exponential growth of the principal due to the accumulation of interest on both the initial principal and the previously earned interest.

Transcript

00:01 In this question we're talking about compounding interest.

00:03 So, converting interest is given on the interest on the principal amount.

00:09 That means every year it is being compounded.

00:11 So the compounding interest is the formula for combating interest is a of t is equal to p multiplied by 1 plus r by n to go out of n.

00:22 Where a of t is the amount that you expect to save after the number of years, yeah, t is taken years and p is the principal amount that here about example if you started the investment with $1 ,000 or $10 ,000 or anything then p means that much amount.

00:39 R means the rate of interest.

00:41 That means it should not be an percentage, it should be explicit in your decimal that is you have to divide the percentage by 100 and then substitute here.

00:49 N means the number of components that is done in and year.

00:54 If it is being compounded annually that means and is a good one if it is being compounded quarterly that means n is equal to 4 and so on so in this question the principal amount is 500 dollars the rate of interest is point somewhere but we have to convert it into you know that is in the decimal to use here and n is it is being said that in this account compounding is done every month that means n is equal to 12 so we need to answer like to the calculations according to the questions that is being asked the first first one we need to express the amount a that is the total a of t in terms of these values that is what we need to do that is on the number of years that is being it is being kept to be in terms of the number of years that has been you know kept in the account so first of all we have this equation and a of t is the amount that we are expecting after the number of years and that is equal to where p is 500, 500 multiplied by 1 plus r is 0 .75 divided by 100 that means 0 .0075 divided by n is 12 right 1 to the power of 12 t is the number of years so this is the equation that represents this particular account or compoting in this particular account so the next thing that we have to do is that we need to find how much will be the amount or how much will be the value of this account after five years ten years 30 years and thirty by years okay so let's start with five years this a of five is equal to 500 multiplied by one plus 0 .0075 divided by two completely divided by 12 to the amount of 12 multiplied by five okay so so that means once we do the calculation we'll get 500 multiplied by 1 plus 0 .0075 divided by 12 right to the power of 60 that means 12.

03:16 That means 500 multiplied by 1 .0 1 .04 so that means that means 500 multiplied by 1 .0 4.

03:24 So that means we'll take 1 .038 because we have to do the rounding off so 1 .038 that is 500 multiple 1 .0338 is 500 multiple 1 .033 is 519 so this will be the amount this much will be the amount after 5 years after 5 years now a of we have to do a next is 10 years 10 years will be 500 multiplied by 1 plus 0 .075 divided by 12 to the bar of 12 multiplied by 10.

04:08 It means 500 multiplied by 1 plus 0 .075 divided by 12, to the bar of 120, which will give us 530.

04:26 To be yours 538 .93 this is the special amount after 10 years and when it comes to a of next one is we have 30 a 30 will be 500 multiplied by 1 plus 0 .0075 divided by 12 to the power of 12 multiplied by 30 right so that will be 12 multiplied by 12 360 right so so the amount will be 626 .126 .1 .2 dollars this will be the amount after 30 years and after 35 years will have 500 multiplied by 1 plus 0 .075 divided by 12 to about 12 multiplied by 35 this will give us total of 650 .650 .040 .04 dollars.

05:41 So this much will be the amount after this many years.

05:43 That is, after five years, we'll have $519.

05:48 So okay.

05:49 So after 10 years, we'll have $538 .93.

05:54 And after 30 years, we'll have $6 .26 .12.

05:58 And after 35 years, we'll have $6 .50 .6 .00.

06:01 Sorry, 04 dollars.

06:03 So, here if you do the whole calculation together just like how i did here you will get you know decimal places and so that you have to round it off to what two decimal places that is to the nearest cent if i just do that one second i'll get it as just yeah this will be around this will be around 519 .1 .1.

06:33 This will be the, you know, approximate value that is being around profit the nearest cent.

06:39 So, so, so near a send means, yeah, 0 .0 .09.

06:43 So that means will be the close approximation compared to this, but yeah, so that is a case.

06:51 Now next moving on to the next question that is the third one that is determine how long it will take for the initial investment to double your answer to the nearest year that's what we need to that is we need a of t to become double the investment that is $1 ,000 a principal amount is $500 right so we need amount with the value to be $1 ,000 which so we need to find out how many years it will take to reach $1 ,000 we know that in 35 years it's only going to reach $650 so that means it's going to take quite a bit of years right we'll see so we need to even deep so that thousand is equal to 500 multiplied by 1 plus rs is 0 .0075 divided by n is 12 multiplied 1 .3 more to the bar of n t so we need to find t right ns 12 12 so this is what we need to find so this will give us 1 plus 0 .0075 by 12 whole to the power of 20 is equal to dividing both sides by 500 we'll get 2 here now that is now applying logarithm here so you can apply a common level of natural whatever you want i'm applying natural load with them so natural law of natural law of doing the calculations inside this will get 1 plus 070 divided by divided by 12 will get 1 .00625 1 .00 yeah 1 .00625 will take 1 .001 so that will be better 12t is equal to natural log of 2 so applying to the properties of logarithin will have 12t multiplied by natural log of 1 .001 is 9199 .99 9 .9 .9.

08:53 Multiplied by 10 to the bar of 9 .995...

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