7.2 Simple and Compound Interest

Interest is the extra amount paid on a loan or earned on savings over time. There are two main types of interest calculations: simple interest and compound interest.

Key Concepts:

• Simple Interest:

  Interest calculated on the original principal amount. \[ I = P \times r \times t \] where:
  • \(I\) = Interest
  • \(P\) = Principal (initial amount)
  • \(r\) = Annual interest rate (as a decimal)
  • \(t\) = Time in years

• Total Amount with Simple Interest:

  \[ A = P + I = P(1 + rt) \]

• Compound Interest:

  Interest calculated on both the principal and previously earned interest. \[ A = P \left(1 + \frac{r}{n} \right)^{nt} \] where:
  • \(A\) = Final amount after interest
  • \(P\) = Principal
  • \(r\) = Annual interest rate (as a decimal)
  • \(n\) = Number of times interest is applied per year
  • \(t\) = Time in years

• Continuous Compound Interest:

  \[ A = P e^{rt} \] where \(e\) is the mathematical constant \(\approx 2.718\).

Examples:

Example 1: Simple Interest Calculation

John deposits \$5000 in a bank account that pays 4\% simple interest per year. Find the total interest earned after 3 years.

Solution:

Step 1: Use the simple interest formula. \[ I = 5000 \times 0.04 \times 3 \] \[ I = 600 \] Step 2: Find the total amount. \[ A = P + I = 5000 + 600 = 5600 \] Thus, John earns **\\(600** in interest, and his total amount after 3 years is **\\)5600**.

Example 2: Compound Interest Calculation

Sarah invests \$2000 in a savings account that offers an annual interest rate of 5\%, compounded yearly. Find the total amount after 4 years.

Solution:

Step 1: Use the compound interest formula. \[ A = 2000 \left(1 + \frac{0.05}{1} \right)^{1 \times 4} \] \[ A = 2000 \left(1.05 \right)^4 \] \[ A = 2000 \times 1.21550625 \] \[ A = 2431.01 \] Thus, Sarah's total amount after 4 years is **\$2431.01**.