Understanding Compound Interest and Simple Interest with Practical Examples

Compound interest and simple interest are two fundamental concepts in finance, often used to calculate the interest on various financial products. This article will explore the relationship between compound interest and simple interest, and demonstrate how to find the rate of interest using practical examples.

Compound Interest vs. Simple Interest

Both compound interest and simple interest are methods to calculate the interest earned on an invested or borrowed amount over a specified period. However, they differ in the way the interest is calculated and added to the principal sum.

Simple Interest is calculated based on the original principal amount. The formula for simple interest is:

SI (P * r * t) / 100

where:

P is the principal amount or initial sum, r is the rate of interest per annum, t is the time in years.

Compound Interest is calculated on the principal amount and also on the accumulated interest of previous periods. The formula for compound interest over 'n' years is:

CI P * (1 r/100)n - P

Practical Example: Finding the Rate of Interest

Let's consider an example where the compound interest for 2 years is 209, and the simple interest for the same period is 200. We will use these values to find the rate of interest.

Step 1: Calculate Daily Simple Interest

Given:

SI for 2 years 200 SI for 1 year 200 / 2 120

We can find the simple interest on 120 for one year by subtracting the simple interest for 2 years from the compound interest for 2 years:

SI on 120 for 1 year 265 - 240 25

Step 2: Calculate the Rate of Interest

The formula to calculate the rate of interest is:

Rate (SI * 100) / (Principal * t)

Here, the principal (P) is 120, and the time (t) is 1 year.

Rate (25 * 100) / (120 * 1) 20.83

Step 3: Confirm the Rate of Interest with Equation-Based Approach

Let's use the formula-based approach to confirm the rate of interest.

Given:

SI for 2 years 200 CI for 2 years 209 SI for 120 for 1 year 25

We start with the simple interest formula:

200 (P * r * 2) / 100

Rearranging gives:

P * r 10000 / r

Next, using the compound interest formula:

209 P * (1 r/100)2 - P

Simplifying, we get:

209 P * (2r/100 r2/10000)

Multiplying both sides by 100:

20900 P * (200r r2)

Using P 10000 / r from equation 1:

20900 * r 10000 * (200r r2)

Rearranging and simplifying:

100r2 - 900r 0

Factoring out r:

r(100r - 900) 0

This gives r 0 or r 9. Therefore, the rate of interest is 9.

Conclusion

In this article, we demonstrated how to find the rate of interest using a combination of simple and compound interest. Understanding these concepts is crucial for making informed financial decisions and managing investments effectively.