What is simple interest and interest compounding? Full explanations with examples and easy-to-use online calculators help you experiment with different parameter values and understand the formulas related to interest compounding. Problems on compound interest with detailed solutions are also included.
What Is Percent Increase?
To understand compounding, you first need to understand the percentage increase of a quantity. If \(P\) is a quantity increased by a percentage rate \(r\), then the new quantity is:
\[ P + rP = P(1 + r) \]Key Idea: A quantity \(P\) increased by a rate \(r\) becomes \(P(1+r)\).
Example 1
\(200\) increased by \(5\%\) becomes:
\[ 200 + \frac{5}{100}\cdot 200 = 200(1+0.05) \]Simple Interest (Not Compounded)
If an amount \(P\) is deposited in a savings account at an interest rate \(r\) that is not compounded, then after \(t\) years the interest earned \(I\) is:
\[ I = Prt \]The total amount \(A\) in the account after \(t\) years is:
\[ A = P + Prt = P(1 + rt) \]Example
\( \$100 \) invested at a \( 3\% \) interest rate (not compounded) for 5 years earns:
\[ 100 \cdot \frac{3}{100} \cdot 5 = 15 \]Yearly Interest Compounding
An amount of money \(P\) (principal) is invested at an annual interest rate \(r\):
- At \(t=0\): \(A=P\)
- At \(t=1\): \(A = P + rP = P(1+r)\)
- At \(t=2\): \(A = P(1+r) + rP(1+r) = P(1+r)^2\)
- At \(t=3\): \(A = P(1+r)^3\)
- At \(t=4\): \(A = P(1+r)^4\)
By extension, after \(t\) years, the total amount compounded yearly is:
\[ A = P(1+r)^t \]Example 2
\( \$1000 \) invested for 3 years at \( 3\% \) compounded yearly:
\[ A = 1000(1+0.03)^3 = 1092.73 \]Interest Compounded n Times Per Year
If the interest is compounded \(n\) times per year, the amount after \(t\) years is:
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]Example 3
\( \$1000 \) invested for 3 years at \(3\%\), compounded twice per year:
\[ A = 1000\left(1+\frac{0.03}{2}\right)^6 = 1093.44 \]Continuous Compounding
Starting from periodic compounding:
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]As \(n \to \infty\), the base expression approaches Euler's constant \(e \approx 2.71828\). Hence, for continuous compounding, the formula becomes:
\[ A = Pe^{rt} \]Example 4
\( \$1000 \) invested for 3 years at \(3\%\), compounded continuously:
\[ A = 1000e^{0.03 \cdot 3} = 1094.17 \]Conclusion: Compare how the same \( \$1000 \) grows under simple interest, annual compounding, and continuous compounding, and determine which method earns the highest return.