At what interest rate (compounded annually) will a sum of 4000 grow to 6000 in 5 years?

Name: Problem 3, Chapter 5: Precalculus
Uploaded: 2020-07-15T18:26:41-08:00
Duration: 4 min 19 s
Channel: Grant Mansfield
Description: At what interest rate (compounded annually) will a sum of 4000 grow to 6000 in 5 years?

At what interest rate (compounded annually) will a sum of ...

Key Concepts

Compounding Frequency

Compounding frequency refers to how often interest is calculated and added to the principal balance of an investment or loan. Annual compounding means that the interest is compounded once per year, a factor that directly influences the growth of the investment as dictated by the compound interest formula.

Exponential Equation Solving

Solving exponential equations involves isolating the exponential term and using logarithmic functions or nth roots to solve for the variable in the exponent. This method is particularly useful in financial problems where one needs to determine the required rate of growth or time period to achieve a desired amount.

Compound Interest

Compound interest is the process of earning interest on both the initial principal and the interest that has been added to the account over time. This concept is essential in finance and economics because it explains how investments grow more rapidly compared to simple interest, where interest is earned only on the principal.

Compound Interest Formula

The compound interest formula, typically expressed as A = P(1 + r)^t, is a fundamental tool for calculating the future value of an investment or loan. In this formula, P represents the principal amount, r is the annual interest rate, and t is the number of periods for which the money is invested or borrowed, reflecting the exponential growth effect of compounding.

Transcript

00:01 All right, so for the question that we're looking at today is we are asked at what interest rate compounded annually will a sum of $4 ,000, i'm just going to write this down, $4 ,000, grow to be $6 ,000 in five years, in five years.

00:28 So what we're first going to use is we're going to use the following formula.

00:32 So it says that a, which is the amount, is equal to the principle of one plus the interest rate to the power of n, or how many years it is.

00:55 So we have our a, which is 6 ,000.

01:02 We have our p, which is 4 ,000.

01:06 We have our n, which is five years.

01:11 We are looking, in this case, for our r, or the interest rate.

01:18 So we're going to put everything into that formula now.

01:21 We have 6 ,000 is equal to 4 ,000, 1 plus r to the power of 5.

01:35 First thing we're going to do is divide both sides by 4 ,000, and that will get us 1 .5 is equal to 1 plus r to the power of 5.

01:59 And next we're going to use the natural logarithm properties for both sides.

02:04 So the lawn of 1 .5 is equal to the lawn of 1 plus r to the power of 5.

02:15 And what we can do now with the one of the properties of the natural logarithm is we can actually take this 5 exponent and bring it out in front of the log.

02:28 So what that is going to look like, it's going to look like this...

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