Understanding the Difference Between SI and CI After Two Years

In financial mathematics, the terms Simple Interest (SI) and Compound Interest (CI) are essential for comprehending how investments grow over time. This article aims to explore the scenario where the difference between Simple and Compound Interest over a period of two years, calculated on a principal sum of Rs. 4000, amounts to Rs. 32. This difference over two years is solely due to the interest on interest characteristic of Compound Interest.

Calculating SI and CI

Simple Interest is calculated using the formula: SI (P * r * t) / 100, where P is the principal, r is the rate of interest (expressed as a percentage), and t is the time in years. Compound Interest, on the other hand, is given by: CI P(1 r/100)^t - P.

The difference between the interest amounts for two years can be calculated using the following formula: SUM D * 100^2 / r^2, where D is the difference in interest between SI and CI, and r is the rate of interest, with t being 2 years in this case.

The Problem

We are given the following data: Principal sum, P Rs. 4000 Difference in interest, SUM Rs. 32 (CI over SI over 2 years) Time, t 2 years

Solving for the Rate of Interest (r)

Substituting these values into the formula, we get:

4000 32 * 100^2 / r^2

Upon rearranging the equation to solve for r^2 and then taking the square root, we obtain:

r^2 80

r √80 ≈ 8.94%

This calculation shows that the rate of interest is approximately 8.94%.

Calculations in Detail

Let's break this down further to understand how the Rs. 32 difference arises:

First Year Simple Interest: SI for the first year (P * r * 1) / 100 4000 * r / 100 40r First Year Compound Interest: CI for the first year P * (1 r/100) - P 4000 * (1 r/100) - 4000 4000 40r - 4000 40r Second Year Interest on Interest: Interest on the second year's compound amount (4000 * (1 r/100) 40r) * r/100 - 40r Simplifying this, we get: 40r * r/100 0.4r^2

Given that the difference in the second year is Rs. 32, we have:

r^2 32 / 0.4 80 r √80 ≈ 8.94%

This calculation confirms the rate of interest to be approximately 8.94%.

Conclusion

Understanding the difference between Simple and Compound Interest over a period of two years is crucial in financial planning and investment analysis. This example clearly illustrates how Compound Interest accounts for the interest on interest, leading to an additional Rs. 32 over the principal sum of Rs. 4000 over two years. This concept is vital for anyone involved in personal finance, investment, and financial planning.