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Apply the Compound Interest Formula for Monthly Compounding Interest
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Introduction

In this lesson, you will learn the difference between simple interest, annual compound interest, and compound interest.


This video illustrates the lesson material below. Watching the video is optional.


Growing From the Simple Interest Formula to the Compound Interest Formula

Simple Interest Formula
The simple interest formula states that the amount of money you have is equal to the initial amount, or the principal amount, plus any new interest:\(A= P + I\). The interest is equal to the principal multiplied by the rate and the time.: \(I = Prt\)

You can put these two equations together so that \(A\) (the amount) is equal to \(P\) (the principal) plus \(I\) (the interest): \(A = P + I\). Instead of using \(I\) for interest you can use what \(I\) is equal to which is \(Prt\):
\begin{align*} A = P + Prt \end{align*}

In the distributive property of multiplication, when something in parentheses is being multiplied, you can either multiply every element within the parentheses individually or multiply after you have completed the operations inside the parentheses. Note the example below where both terms are multiplied by \(3\):

\begin{align*} &3\cdot 2 + 3\cdot 4 &\color{red}\small\text{Given expression to factorize}\\\\ &3 (2 + 4) &\color{red}\small\text{Factorize a common factor: \(3\)}\\\\ \end{align*}

The common factor will be outside the parentheses and the remaining factors inside the parentheses: \(3(2+ 4)\)

With that in mind, consider the simple interest equation: \(A = P + Prt\).

On the right-hand side, there is a P in both terms that are part of the addition problem.

\begin{align*}
A&=P+ Prt &\color{red}\small\text{Given formula to factorize}\\\\
A&=P(1) + Prt &\color{red}\small\text{Note that P is the same as \(P\times 1\)}\\\\
A& = P(1 + rt) &\color{red}\small\text{Factorize a common factor: \(P\)}\\\\
\end{align*}

This is the simple interest formula that you will see used in the real world and is the first level of interest formulas.

Annual Compound Interest Formula
The next formula, annual compound interest, is the second level of interest formulas:

\begin{align*} A = P(1+r)^t\end{align*}

Annual compound interest can do more because it quickly calculates compound interest using the t in the exponent position, while the simple interest formula can only compound by using the same formula repeatedly. The annual compound interest formula can calculate the compound interest, but only when t is in years.

Compound Interest Formula
The third level of interest formulas is the compound interest formula. It can do so much more than either of the previous two formulas. The only difference between the compound interest formula and the annual compound interest formula is that the compound interest has the variable n, which stands for the number of compounding periods per year. It uses n to divide r, and multiplies the t in the exponent position by n.

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.

Here is the compound interest formula:

\begin{align*} \large A = P ( 1 + {\frac {r}{n}} ) {^n}{^t} \end{align*}

\(A\) = Amount (ending amount)

\(P\) = Principal (beginning amount)

\(r\) = Rate (annually) as a decimal

\(t\) = Time in years

\(n\) = Number of compounding periods per year

Calculating Compound Interest Using the Compound Interest Formula


The following 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.

Example 1
Calculate the end amount where the principal is $300, the rate is 8% compounded monthly (12 times per year), and the time is 3 years.

Substitute these numbers into the equation. Remember, the rate has to be expressed as a decimal when using these formulas, so 8% is 0.08. Follow the order of operations and use all the digits in the calculator from each previous step.

\begin{align*}& P = 300, \;r=0.08, \; t= 3\;years, \;n=12 &\color{red}\small\text{Label the given information}\\\\
&A= P(1 + \frac{r}{n})^{nt} &\color{red}\small\text{Use the compound interest formula}\\\\
& A=300(1+\frac{.08}{12})^{12\times3} &\color{red}\small\text{Substitute the given info to the formula}\\\\ &A=300(1+0.00667)^{12\times3} &\color{red}\small\text{Simplify the parenthesis first by dividing}\\\\ &A=300(1.00667)^{12\times3} &\color{red}\small\text{Continue simplifying the parenthesis by adding}\\\\ &A=300(1.00667)^{36} &\color{red}\small\text{Next multiply the exponents}\\\\ &A=300(1.2704) &\color{red}\small\text{Calculate the power}\\\\ &A=381.12 &\color{red}\small\text{Multiply last}\\\\ \end{align*}

Avoid rounding errors by not rounding until the final answer.


Things to Remember


  • A compound interest formula is useful when the interest is compounded more than once a year.
  • The formula for compound interest is: \(A=P(1+\frac{r}{n})^{nt}\).

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 you 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, you are finding the ending balance of an account that pays 8% annual interest compounded monthly, so you 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 you need the following:

    Principal (beginning amount) = \(P\) = \({\color{MediumVioletRed}2000}\)

    Rate (annually) as a decimal = \(r\) = \({\color{DodgerBlue}0.08}\)

    Time in years = \(t\) = \({\color{MediumSeaGreen}3}\)

    Number of compounding periods per year = \(n\) = \({\color{DarkOrange}12}\) (since the account is compounded monthly)

    Input each of the above into the 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 \(1.270237\):

    \(\text{A}={\color{DodgerBlue}2540.47}\)

    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:
    You started with \($2,000\) and at the end of 3 years you had \($2,540.47\). The difference between the two is \($540.47\) so that is how much you 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 you double the interest earned at \(4\%\) you get \($254.54 \times 2 = $509.08\). At \(8\%\), you 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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