Introduction

The concept of interest is fundamental in finance and plays a significant role in various financial transactions. Among the two primary types of interest, simple interest and compound interest, the difference of 2 rupees when invested for 2 years at a 5% annual interest rate becomes a unique problem. Let's explore this scenario in greater detail.

The Problem

The problem states that the difference between the compound interest (CI) and simple interest (SI) on a sum for 2 years at an annual rate of 5% is 2 rupees. The question is, what is the principal amount (P)? To solve this, we need to understand the basic formulas and the subtle differences between simple and compound interest.

For simplicity, let's assume that the interest is compounded annually.

Calculating Simple Interest (SI)

Simple interest is straightforward and is given by the formula:

SI P × R × T / 100

where P is the principal amount, R is the rate of interest, and T is the time period in years.

Calculating Compound Interest (CI)

Compound interest, on the other hand, is calculated based on the accumulated interest of the prior periods. The formula for compound interest is:

CI P × (1 R/100)T - P

Let's break this problem down step by step to find the principal amount.

Step 1: Calculate SI for the First Year

For the first year, the SI is:

SI1st year P × 5/100 0.05P

Step 2: Calculate CI for the First Year

In the first year, the CI is the same as SI because the interest is compounded based on the principal amount:

CI1st year 0.05P

Step 3: Calculate CI for the Second Year

For the second year, the interest is calculated on the amount after the first year, which is P 0.05P:

Interest for the second year (P 0.05P) × 5/100 0.05P 0.05 × 0.05P 0.05P 0.0025P 0.0525P

Step 4: Calculate the Difference Between CI and SI for the Second Year

The difference between CI and SI for the second year is:

Difference 0.0525P - 0.05P 0.0025P

Step 5: Set Up the Equation

According to the problem, this difference is 2 rupees:

0.0025P 2

Solving for P:

P 2 / 0.0025 800 rupees

Conclusion

The principal amount (P) is 800 rupees. This problem helps illustrate the subtle difference in how compound and simple interest work over multiple periods.

Formula Recap:

- Simple Interest: SI P × R × T / 100

- Compound Interest: CI P × (1 R/100)T - P

Key Takeaways:

1. **Understanding Simple Interest (SI)**: SI is calculated based on the original principal amount.

2. **Understanding Compound Interest (CI)**: CI is calculated based on the accumulated interest of the prior periods.

3. **The Difference Between SI and CI**: For a 5% annual interest rate and 2 years, the difference between CI and SI is given by 0.0025P, where P is the principal amount.