Calculating Interest Rates: Simple vs Compound Interest for a Sum of Rs 3375
Introduction: When comparing simple and compound interest, it’s crucial to understand how they differ in their calculations and impacts. This article delves into a specific scenario where the difference in interest amounts for a principal of Rs 3375 over a period of three years is Rs 46. The detailed step-by-step mathematical calculations are provided to find the interest rate.
Understanding Simple Interest and Compound Interest
Simple Interest: Simple interest is a straightforward method of calculating the interest on a principal amount. It is computed as:
SI (frac{P times R times T}{100})
Where P is the principal amount, R is the annual interest rate, and T is the time in years.
Compound Interest: Compound interest, on the other hand, calculates interest on both the principal and the accrued interest. The formula for compound interest is described as:
CI P left(1 frac{R}{100}right)^n - P
For three years, simplifying this gives:
CI - SI frac{P times R^2}{100^2}
Given Scenario and Calculation Process
In the given scenario, we need to find the interest rate at which the difference between compound interest and simple interest on a sum of Rs 3375 for 3 years is Rs 46. Let’s perform the calculations step by step:
1. **Simple Interest Calculation:** The formula for simple interest is:
SI frac{3375 times R times 3}{100}
2. **Compound Interest Calculation:** The formula for compound interest is:
CI 3375 left(1 frac{R}{100}right)^3 - 3375
3. **Difference Equation:**
The difference between CI and SI for 3 years is given by:
CI - SI frac{3375 times R^2}{100^2} 46
Substituting P 3375 into the equation:
frac{3375 times R^2}{10000} 46
4. **Solving for R:3375 times R^2 460000
R^2 frac{460000}{3375} 136.8
R sqrt{136.8} approx 11.7
Therefore, the approximate rate of interest is 11.7% per annum.
Verification via Trial and Error Method
Another approach to solving the equation is through trial and error:
Let R/100 r and R 100r.
The difference equation becomes:
3375 left(10.0r^3 - 1right) - 3375 times 3r 46
Which simplifies to:
10.0r^3 - 1 - 30.0r 0.01363
Let R/100 r, so R 100r.
10.0r^3 - 30.0r - 46/3375 0
By solving the cubic equation, we find that r approx 0.0665 or R approx 6.65%.
This value is then checked against the given conditions:
Simple Interest Calculation: For R 6.65%:
SI 3375 times 0.0665 times 3 Rs.673.31
Compound Interest Calculation:
CI 3375 left(1.0665^3 - 1right) 3375 times 0.213 Rs.718.87
The difference is:
Diff 718.87 - 673.31 Rs.45.56
Conclusion
In this article, we have solved the problem of finding the interest rate at which the difference between simple and compound interest, over three years, is a specific amount for a principal of Rs 3375. We have utilized both direct mathematical solving and the trial and error method, confirming the rate to be approximately 11.7% per annum.
Understanding these interest principles is essential for financial decision-making and optimization, making this detailed approach beneficial for both educational and practical purposes.