Question

Tutorial on The Compound Interest Formula

This can be used to find things like the total money in an account or owed after interest, the principal or initial amount invested/borrowed, and the total interest. You may also see questions in the form of "Which of the formulas" that will need a partial solution to get the correct answer.

Answer 1

Nice Job Eric!

Answer 2

Gonna fix some ( ) size issues...

Answer 3

Whoa nice tutorial :)

Answer 4

The Compound Interest Formula

\(\Large A = P\left(1+\dfrac{r}{n}\right)^{nt}\)

In this formula:

t = time in years
A = amount of money + interest after t years
P = principal/initial amount
r = annual rate of interest (APR) as a decimal
n = number of times the interest is compounded per year

This can be used many ways. For example, what if you had $1000 and wanted to know how much it would become in a monthly compounding account after 5 years. Say the APR was 2%.

First, convert 2% into a decimal. Percent comes from per centum which is Latin for "divide by 100." So: \(\Large \dfrac{2}{100}=.02\)

Now you can put the numbers into the formula:

\(\Large A = P\left(1+\dfrac{r}{n}\right)^{nt}\) becomes \(\Large A = 1000\left(1+\dfrac{.02}{12}\right)^{(12)(5)}\)

\(\Large A = 1000\left(1+\dfrac{.02}{12}\right)^{60}\)

\(\Large A = 1000(1+0.00167)^{60}\)

\(\Large A = 1000(1.1052996)\)

\(\Large A = 1105.30\)

But what if I just wanted to know how much interest was made? Well, the interest is the total amount A minus the principal P.

\(\Large I = A-P\)

\(\Large I = 1105.30-1000\)

\(\Large I = 105.30\)

Also, this can be solved for a number of other things. For example, if I said a person had $1105.30 in their account that has been compounding monthly for 5 years at an APR of 2%, you could use this information to work back to the Principal.

\(\Large $1105.30 = P\left(1+\dfrac{.02}{12}\right)^{60}\)

\(\Large $1105.30 = P(1+0.00167)^{60}\)

\(\Large $1105.30 = P(1.1052996)\)

\(\Large $1105.30/1.1052996 = P\)

In the end I get 1000.0003618928297811742626162174 due to rounding errors, but that is still $1000 as the principal.

As you can see, algebra and the Compound Interest Formula lets you easily find the interest earned, total amount in an account, or the principal. With a little more work you could also find the rate, compounding times, or number of years, but those are not as simple as those first three.

I have also seen questions where they do not want the complete solution. For example, they might say:

You put $1000 in a savings account that earns 2% interest compounded monthly. You want to keep the money in the account for 5 years. Which of the formulas below can help you calculate how much money will be in the account at the end of 5 years?

\(A = 1000(1+0.00167)^{60}\)

\(A = 1000(1+0.00334)^{30}\)

\(A = 1000(1+0.0167)^{60}\)

\(A = 1000(1+0.00167)^{120}\)

As you look at my calculations above, you can see the first one is right. However, how can you get to that point rapidly?

Well, I would start by just doing the number of compoundings times the time, nt, in the exponent. First, nt is an easy calculation as it is usually multiplication of two whole numbers. Second, due to how they ask a lot of these questions, you generally eliminate half the possibilities or find the answer with just that one calculation.

If doing rt does not finish it, then find the rate as a decimal and, if needed, divide it by n. Because you may end up with a repeating fraction, this can be a little messier of an answer. That is why I suggest leaving it for second. However, if you do need to take this second step, you should be able to eliminate any remaining wrong possibilities and have the answer to a "Which of the formulas" question.

Answer 5

There it is with the ( and ) size mathing the fractions.

Answer 6

Calculating 'r' when the other variables are given is another possibility to consider.

And calculating 'n' too.

Answer 7

Yes, I mention that. They just tend to take a little more work.