The Difference Between Simple and Compound Interest on a Specific Sum of Money for Two Years at 4 Percent Annum is Rs. 20. What Is the Sum?

Understanding the difference between simple and compound interest is vital in the fields of finance and economics. In this article, we will delve into the mathematical principles behind simple and compound interest, providing clear explanations and solving relevant problems. We will explore how the difference between the two types of interest can be used to determine the initial sum of money.

Simple Interest Calculation

Simple interest is calculated using a straightforward formula:

Simple Interest (SI) PTR / 100

Example: Finding the Sum of Money

Let's assume the principal (P) is Rs. 100, the rate (R) is 5 percent, and the time (T) is 2 years.

Simple Interest (SI) 100 x 5 x 2 / 100 10

Compound Interest Calculation

Compound interest is more complex as it involves interest being calculated on both the principal and the accumulated interest over time.

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

Example: Finding the Compound Interest

Using the same principal (P) of Rs. 100, the rate (R) of 5 percent, and the time (T) of 2 years, we can calculate the compound interest as follows:

Compound Interest (CI) 100(1 5/100)2 - 100

CI 100(1.05)2 - 100

CI 100 x 1.1025 - 100

CI 110.25 - 100

CI 10.25

Determining the Initial Sum of Money

By subtracting the simple interest from the compound interest, we can find the difference:

Difference (DI) CI - SI 10.25 - 10 0.25

When P is 100, the difference is 0.25, thus:

When the difference is 60, the sum can be calculated as:

Sum (P) 60 x 100 / 0.25 24000

Additional Example and Formula Application

Let's work through another example with a slightly different rate of interest (5 percent) and time period (2 years).

Example: Finding the Sum of Money

Let the sum (P) be Rs 100, the rate of interest (R) be 5 percent, and the time (T) be 2 years.

Simple Interest (SI) 100 x 5 x 2 / 100 10

Compound Interest (CI) 100(1 5/100)2 - 100 100(1.1025) - 100

CI 110.25 - 100 10.25

Now, the difference between CI and SI in 2 years is:

CI - SI 10.25 - 10 0.25

Therefore, if the difference is 60, the sum can be found using the formula:

Sum (P) 60 x 100 / 0.25 24000

Formulas and General Summary

The formulas for simple and compound interest can be generalized as:

The difference between compound and simple interest for two years can be calculated using the formula:

Difference (D) (P x R2) / 10000

Using this formula, we can determine the principal amount (P) as follows:

P (D x 10000) / (R2)

In the given problem, the difference is 20, and the rate of interest is 4 percent. Substituting these values, we get:

Difference (D) 20 (P x 42) / 10000

Solving this equation, we find:

P (20 x 10000) / (42) 12500

Thus, the sum is Rs. 12500.