Simple and Compound Interest Calculation: A Complex Problem Solved
The relationship between simple interest and compound interest presents a fascinating topic in mathematical finance. This article provides a step-by-step solution to a complex problem where a sum of money yields a specific interest over a period of time. By understanding the principles of simple and compound interest, we can effectively solve such questions and gain valuable insights into financial mathematics.
Understanding Simple and Compound Interest
Before diving into the problem, it is essential to understand the basic principles of simple interest and compound interest.
Simple Interest (SI)
Simple interest is calculated using the formula:
SI P × R × T
Where:
P is the principal amount, R is the rate of interest per period, T is the time period in years.It is a straightforward method of calculating interest and does not take into account the interest on the interest.
Compound Interest (CI)
Compound interest, on the other hand, is the interest calculated on the initial principal and also on the accumulated interest of previous periods. The formula for compound interest is:
CI P{1 (R/100)}^n - P
Where:
P is the principal amount, R is the rate of interest per period, n is the number of compounding periods.The Problem at Hand
A certain sum of money, when invested, yields a simple interest of Rs 225 over three years, and a compound interest of Rs 153 over the same period. The unique aspect of this problem is that the compound interest compounded annually and the simple interest in the first year are the same.
Solving for the Principal Amount (P)
Given:
Simple interest for 3 years Rs 225, Compound interest for 2 years Rs 153.Let's start by breaking down the problem into smaller parts:
Step 1: Simple Interest for the First Year
The simple interest for the first year is calculated as:
SI_1 225 / 3 Rs 75
Step 2: Simple Interest for the Second Year
The simple interest for the second year, considering the interest from the first year, is:
SI_2 153 - 75 Rs 78
Step 3: Principal Amount (P) for the First Year
The principal amount (P) for the first year is:
SI_1 P × R × T
Rearranging to find P:
P SI_1 / (R × T)
Given that P 80 and T 1 year, we can find R:
R SI_1 × 100 / (P × T) 75 × 100 / (80 × 1) 7500 / 80 93.75%
Step 4: Compound Interest for 2 Years
The compound interest for 2 years is given as Rs 153. Using the compound interest formula:
CI P{1 (R/100)}^n - P
Substituting the known values:
153 P{1 (93.75 / 100)}^2 - P
Since the compound interest for 2 years should be equal to Rs 153, we simplify this to find P:
P 1875
Conclusion
The principal amount (P) for the investment is Rs 1875. This solution demonstrates the application of both simple and compound interest principles and highlights the importance of understanding the underlying mathematical concepts in financial mathematics.
Key Takeaways:
Simple interest and compound interest have distinct formulas and applications. Understanding the relationship between simple interest in the first year and compound interest can provide a shortcut in solving complex problems. The rate of interest can be derived from the provided interest amounts and the time period.Keywords: simple interest, compound interest, financial mathematics