COMPOUND INTEREST In Exercises 25 and 26, determine the time necessary for 1000 to double if it

COMPOUND INTEREST In Exercises 25 and 26, determine the time...

Key Concepts

Compound Interest

Compound interest is the process where interest is calculated on both the initial principal and the accumulated interest from previous periods. This mechanism causes the investment to grow at an exponential rate, distinguishing it from simple interest where only the principal earns interest.

Compounding Frequency

Compounding frequency refers to the number of times interest is calculated and added to the principal in a given time period, such as annually, monthly, or daily. Changing the frequency affects the total amount accrued by the end of the period even if the nominal annual interest rate remains the same.

Continuous Compounding

Continuous compounding is an idealized version of compounding where interest is added at every possible instant, leading to an exponential growth that can be modeled mathematically using the natural exponential function. This represents the limit of the compound interest formula as the compounding frequency approaches infinity.

Doubling Time

Doubling time is the length of time required for an investment to grow to twice its initial size. It is typically calculated by setting the compound interest formula equal to two times the initial principal and solving for the time variable, often involving logarithmic functions to isolate the variable.

Exponential Growth

Exponential growth describes the process in which a quantity increases at a rate proportional to its current value. In finance, this concept is fundamental to understanding how investments grow over time through compound interest, as the growth accelerates over successive periods.

Transcript

00:01 If we have our formula for compound interest and for compounding continuously, where a is our final amount, p is our initial principal amount, rate as a decimal, n is the number of times compounded per year, and t is in years.

00:25 And now we specifically have a rate of 6 .5%, which we want to think of as 0 .0 .0 .0 .2.

00:32 065 as a decimal.

00:35 For part a we want to know how long would it take a thousand dollars a double if it was compounded annually? well, we want our final amount then to be $2 ,000 if we're starting with a thousand and n is equal to one in this case and so it'd be one plus the rate as a decimal over 1 times one t times t or solving for t how long is this going to take? divide both sides by a thousand and you get two equals and add 1 plus 0 .065, you get 1 .065 to the 1 times t is t.

01:10 Well, one way to solve this is with logarithms.

01:14 And so we could take the log of both sides.

01:17 The log of 2 would equal the log of 1 .065 to the t.

01:29 And the power rule says we can bring the t out front, and t would then be equal to log of 2 over log of 1 .065.

01:42 And that means t is about, let's calculate it, log of 2 divided by log of 1 .065.

02:02 And that is about 11 .007 years.

02:12 That's part a.

02:14 Part b, monthly.

02:15 Well, the only difference here is that n is 12 instead of 1.

02:21 So 2 ,000 equals 1 ,000 times 1 plus 0 .065 over 12 to the 12t.

02:33 Divide both sides by 1 ,000, we get 2 equals 1 plus 0 .065 divided by 12 is 1 .0054 to the 12t.

03:04 Next, take the log of both sides.

03:08 So the log of 2 would equal the log of 1 .0054 to the 12t.

03:19 And now the power rule says that the 12t can be moved out front.

03:26 So it's 12t times the log of 1 .0054.

03:30 So we have log of 2 divided by 12 times.

03:38 The log of 1 .0054 would be equal to t to get this is a one product so you can divide by both at once which means t would be approximately we go to a calculator here 12 divided by excuse me not 12 log of 2 divided by 12 divided by 12 log of 1 .0054 and that is about 10 .726 but better yet on this if we don't round our 1 plus 0 .065 over 12 and we keep it as is will be more accurate and so a better answer would be about 10 .693 so i'm going to replace that and i don't know why i said 12 to begin with because that's that's not what it was, but that's okay...

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