Understanding Compound Interest: Calculating the Principal and Interest Rate

Compound interest is an essential concept in finance that helps determine the interest earned on an initial principal amount over multiple periods.

Introduction to Compound Interest

Compound interest is calculated not only on the principal amount but also on the accumulated interest from previous periods. This method of calculating interest can be particularly useful in financial planning and investments. The formula for compound interest is given by:

Formula for Compound Interest

A P (1 r)n

Where:

A is the amount of money accumulated after n years including interest. P is the principal amount (initial amount of money). r is the annual interest rate (as a decimal). n is the number of years the money is invested or borrowed for.

Problem and Solution

In this problem, we are given that the compound interest for two years is 5400 and for three years is 6480. We need to determine the principal amount and the interest rate per year.

Step-by-Step Solution

1. **Define the known values**:

Compound interest for 2 years: 5400 Compound interest for 3 years: 6480

2. **Express the amount after 2 years and 3 years**:

A2 P 5400 A3 P 6480

3. **Relate the amount after 3 years to the amount after 2 years**:

A3 A2(1 r)

Therefore,

P 6480 (P 5400)(1 r)

4. **Find the interest for the third year**:

Interest for the third year A3 - A2 6480 - 5400 1080

5. **Express the interest for the third year using the amount after 2 years**:

1080 (P 5400) * r

6. **Solve for r using the difference of known values**:

r 1080 / (P 5400)

7. **Substitute r back into the equation**:

P 6480 (P 5400) * (1 1080 / (P 5400))

8. **Simplify the equation to solve for P**:

P 6480 P 5400 1080

9. **Calculate the principal amount P**:

P 6480 - P - 5400 1080 1080

P 5400 * r

10. **Substitute r back into the equation to find P**:

P 5400 * (1080 / (P 5400))

11. **Derive the principal amount**:

P 5400 * 0.25 13500

12. **Calculate the interest rate r**:

r 1080 / (13500 5400) 0.08 8%

Conclusion

The sum of the compound interest for two years and three years is 11880, which includes the given values of 5400 and 6480.

Final Answer

The principal amount is 13500 and the interest rate per year is 8%.

Key Takeaways

Compound interest helps calculate interest earned over multiple periods. The formula for compound interest is essential in financial planning. By solving for the principal amount and interest rate, we can accurately determine investment growth.