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.

Video Source (07:57 mins) | Transcript

Practice Problems

1. 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

   x

   Solution:
   \(1,000(1+\frac{0.09}{12})^{(12\times1)}=1,000(1.0075)^{12}=\$1,093.81\)
   )
2. 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

   x

   Solution:
   \($2254.54\)
   )
3. Look at the answer to question 2. How much interest was earned over the three years? (
   Solution

   x

   Solution:
   \($254.54\)
   )
4. 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

   x

   Solution:
   \($2540.47\)
   Details:
   In this situation, we are finding the ending balance of an account that pays 8% annual interest compounded monthly, so we will use the formula:
   
   \({\text{A}}={\text{P}}\left(1+\dfrac{\text{r}}{\text{n}}\right)^{\text{nt}}\)
   
   To find the amount in the account at the end of three years we need the following:
   
   P = Principal (beginning amount) = \({\color{MediumVioletRed}2000}\)
   
   r = Rate (annually) as a decimal = \({\color{DodgerBlue}0.08}\)
   
   t = time in years = \({\color{MediumSeaGreen}3}\)
   
   n = number of compounding periods per year = \({\color{DarkOrange}12}\) (since the account is compounded monthly.
   
   We plug each of the above into our formula:
   
   \(\text{A}={\color{MediumVioletRed} 2000}\left ( 1+\dfrac{{\color{DodgerBlue} 0.08}}{{\color{DarkOrange} 12}} \right )^{ {\color{DarkOrange} 12}\cdot {\color{MediumSeaGreen} 3}}\)
   
   Then divide \({\color{DodgerBlue}0.08}\) by \({\color{DarkOrange}12}\) to get:
   
   \(\text{A}=2000(1 + {\color{DodgerBlue}{0.0066667}})^{12\cdot3}\)
   
   Then add 1 to \(0.006667\):
   
   \({\text{A}}=2000({\color{DodgerBlue}{1.0066667}})^{12\cdot3}\)
   
   Then multiply 12 by 3:
   
   \({\text{A}}=2000(1.0066667)^{\color{DodgerBlue}{36}}\)
   
   Then raise \(1.0066667\) to the thirty-sixth power:
   
   \(\text{A}=2000\left ({\color{DodgerBlue} 1.270237}\right )\)
   
   Then multiply 2000 by \(1.270237\):
   
   A = \({\color{DodgerBlue}2540.47}\)
   
   So the account will have \($2,540.47\) at the end of 3 years
   )
5. Look at the answer to question 4. How much interest was earned over the three years? (
   Solution

   x

   Solution:
   \($540.47\)
   Details:
   We started with \($2,000\) and at the end of 3 years we had \($2,540.47\).The difference between the two is \($540.47\) so that is how much we earned in interest.
   )
6. 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

   x

   Solution:
   More than doubled.
   Details:
   If we double the interest earned at \(4\%\) we get \($254.54 \times 2 = $509.08\). At \(8\%\), we earn \($540.47\) in interest. The interest at \(8\%\) is more than double the amount earned at \(4\%\).
   )
   1. Exactly the same
   2. Less than doubled
   3. Exactly doubled
   4. More than doubled

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