In the real world, interest is often compounded more than once a year. In many cases, it is compounded monthly, which means that the interest is added back to the principal each month.

In order to calculate compounding more than one time a year, we use the following formula:

\(\displaystyle A = P ( 1 + {\frac {r}{n}} ) {^n}{^t} \)

A = Amount (ending amount)

P = Principal (beginning amount)

r = Rate (annually) as a decimal

t = Time in years

n = Number of compounding periods per year

The following video will explain a little more about each of the formulas we have learned to calculate interest so far and how they are related.

Video Source (05:01 mins) | Transcript

The following video demonstrates how to do the compound interest calculation using the order of operations. It also demonstrates how to enter the numbers into a calculator in order to avoid rounding errors.

## How to Avoid Rounding Errors

Avoid rounding errors by **not** rounding until the final answer. Do this by following the order of operations and using all the digits in the calculator from each previous step.

### Practice Problems

- If you invest \($1000\) in an account that pays \(9\%\) interest annually, compounded monthly, what is the total amount of money that you would have at the end of one year? (Hint: In this case, \(n = 12\) because it is compounding monthly, but \(t = 1\) because we are calculating for 1 year.) (Solution
- Suppose that you invest \($2000\) in an account that pays \(4\%\) interest annually, compounded monthly. How much money would you have in the account after three years? (Hint: \(n = 12\) and \(t = 3\).) (Solution
- Look at the answer to question 2. How much interest was earned over the three years? (Solution
- Suppose that you invest \($2000\) in an account that pays \(8\%\) interest annually, compounded monthly. How much money would you have in the account after three years? (Solution
- Look at the answer to question 4. How much interest was earned over the three years? (Solution
- Compare the answers to questions 3 and 5. When the interest rate is doubled from \(4\%\) to \(8\%\), What happens to the amount of compound interest earned after three years? (Solution
- Exactly the same
- Less than doubled
- Exactly doubled
- More than doubled

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