Understanding Compound Interest and the Rate of Interest

Compound interest is a powerful financial concept that often plays a vital role in investment growth. The rate of interest is the percentage by which the principal increases over a specified period. This article dives into how to calculate the rate of interest when given the compound interest over different periods. We will use a specific example to illustrate the process.

The Example Problem

A sum of money was invested at a compound interest rate. After 2 years, the interest earned was Rs. 4200, and after 4 years, the interest earned was Rs. 9282. Our goal is to determine the rate of interest based on this information.

Using the Compound Interest Formula

The formula for compound interest is:

A P(1 r)^n

Where:

A is the amount after n years, P is the principal amount (initial investment), r is the rate of interest (as a decimal), n is the number of years.

Setting Up Equations

We are given:

The amount after 2 years, A_2 4200, which means the interest earned is Rs. 4200, The amount after 4 years, A_4 9282, which means the interest earned is Rs. 9282.

We can set up the following equations:

A_2 P(1 r)^2 and A_4 P(1 r)^4

Expressing A_4 in Terms of A_2

Next, we express A_4 in terms of A_2.

A_4 P(1 r)^4 P(1 r)^2 cdot (1 r)^2 A_2 cdot (1 r)^2

Given that:

9282 A_2 cdot (1 r)^2

Solving for 1 r^2

We can solve for 1 r^2 by dividing 9282 by 4200:

1 r^2 frac{9282}{4200} approx 2.20857142857

Finding r

To find r, we take the square root:

sqrt{1 r^2} approx sqrt{2.20857142857} approx 1.485

Then, we solve for r:

r approx 1.485 - 1 approx 0.485

To express r as a percentage:

r approx 0.485 times 100 approx 48.5%

Thus, the rate of interest is approximately 48.5%.

Example with Known PV and Interest

To further illustrate, let's consider a scenario where the interest for two years is Rs. 4200 and the present value (PV) is Rs. 20,000. Using the compound interest formula:

20000 times (1 i)^2 29282

Calculating the future value (FV), we get:

1.4642 times 20000 29282

Interest FV - PV

Rs. 29282 - Rs. 20000 Rs. 9282

Conclusion

Understanding the rate of interest and compound interest is crucial for making informed investment decisions. By applying the compound interest formula, we can calculate the rate of interest based on the given interest amounts over different periods.

Keywords: compound interest, rate of interest, compound interest formula