## January 09, 2012

### Compound interest

Photo courtesy of Mooster
Someone sent me in a question concerning compound interest...
Please send me the formula for compound interest and explain line by line.

p(1 + rate)3

What does the 1 stand for and must you add it to the rate of say 10%?

100(1+10)3 years = ???

Obviously I am looking for a basic course?

The formula that this person is using is correct... the formula for compound interest is

A = p(1 + r)t

but this formula doesn't give us the amount of interest -- it gives us the amount of money you would withdraw after t years. In the formula, p is the original principal, r is the interest rate, and t is the time in years.

However, we cannot put the interest rate in as he did. If r = 10%, then r = 0.1 must be used in this formula. In other words, FIRST convert your percentage into a decimal.

For example, if the principal is $5000 and r = 10% = 0.1, then we get A =$5000 × 1.1t

The number 1 in the formula p(1 + r)t doesn't stand for anything by itself. It comes from using the distributive property in simplifying p + pr into p(1 + r).

Which, p + pr is the principal + interest earned after one year; that is what you could withdraw after one year.

BUT if you don't withdraw it, but leave it all to earn more interest, then that becomes your "new" principal, and the year after that you will have that times 1 + r.

Basically, this is how compound interest works:

After year 1: You have p + pr which is p(1 + r). Notice that your original principal got multiplied by (1 + r). If you leave this on the account to earn more interest, then next year you have that amount times (1 + r).

After year 2: you have p(1 + r)(1 + r). If you leave this on the account to earn more interest, then next year you have that amount times (1 + r).

After year 3: you have p(1 + r)(1 + r)(1 + r). Let's simplify this using an exponent: You have p(1 + r)3

After year 4: you have p(1 + r)4

And so on. After year t, you have p(1 + r)t.
Hope this helps some!