Calculating Principal Amount from Interest Differences

In finance and economics, understanding the difference between compound interest and simple interest is crucial. This article will guide you through the process of calculating the principal amount by leveraging the difference between compound and simple interests over a specific time period. We’ll walk through examples and provide a detailed solution with step-by-step instructions.

Understanding Compound and Simple Interest

Before we dive into the problem, let's review the formulas for simple and compound interest:

Simple Interest (SI)

The formula for simple interest is:

SI P × r × t

Where:

P is the principal amount (the initial sum of money) r is the annual interest rate t is the time the money is invested or borrowed for, in years

Compound Interest (CI)

The formula for compound interest is:

CI P × (1 r)^t - P

This can be simplified as:

CI P × [(1 r)^t - 1]

Problem Solving

Given: Rate (r) 35% 0.35, Time (t) 3 years, Difference between CI and SI 400000.

Step 1: Calculate Simple Interest (SI)

Using the simple interest formula:

SI P × 0.35 × 3 1.05P

Step 2: Calculate Compound Interest (CI)

Using the compound interest formula:

CI P × [1 0.35^3 - 1]

First, we need to calculate 1 0.35^3.

1.35^3 1.35 × 1.35 × 1.35 2.460375

Therefore:

CI P × (2.460375 - 1) 1.460375P - P

So:

CI 1.460375P - P 1.460375P - 1P 0.460375P

Step 3: Find the Difference Between CI and SI

The difference between CI and SI is:

CI - SI 0.460375P - 1.05P

Combine like terms:

CI - SI (0.460375 - 1.05)P -0.589625P

Step 4: Set the Difference Equal to 400000

We know the difference is 400000, so:

-0.589625P 400000

Step 5: Solve for P

Rewriting and solving for P:

P frac{400000}{0.589625} approx 676267.24

Thus, the principal amount is approximately 676,267.24.

Conclusion

The principal amount can be calculated using the difference between compound and simple interests over a specified time period. In this example, the principal amount that would yield a difference of 400000 between CI and SI over 3 years at a rate of 35% is approximately 676,267.24.