Understanding the Relationship Between Compound Interest (CI) and Simple Interest (SI) in Financial Calculations

When dealing with financial calculations, it is important to understand the relationship between compound interest (CI) and simple interest (SI). This relationship is crucial when analyzing the growth of a principal amount over different time periods. In this article, we will explore how to determine the rate of interest given a specific ratio of the difference between CI and SI for different time periods.

The Concept

Simple interest and compound interest are two different ways to calculate the interest earned on a principal amount over a certain period. Simple interest is calculated only on the principal amount, while compound interest is calculated on the principal as well as the accumulated interest of previous periods.

Simple Interest (SI)

Simple interest is calculated using the formula:

SI P * R * T / 100

Compound Interest (CI)

Compound interest is calculated using the formula:

CI P * (1 R/100)T - P

Problem Statement

The problem involves finding the ratio of the total interest earned over 3 years to the total interest earned over 2 years, given a specific ratio of the difference between the CI and SI for 3 years and 2 years.

Let P denote the common principal amount, and R denote the common annual percentage rate of interest. The amounts of compound interest (Cn) and simple interest (Sn) for 2 and 3 years are given as:

C2 P[1 R/100^2 - 1], S2 P[2R/100]

C3 P[1 R/100^3 - 1], S3 P[3R/100]

Solving the Problem

The given ratio is:

C3 - S3 : C2 - S2 29:9

We can express C3, S3, C2, and S2 in terms of P and R:

C3 - S3 P[1 R/100^3 - 1 - 3R/100]

C2 - S2 P[1 R/100^2 - 1 - 2R/100]

Setting up the ratio equation and simplifying:

(P[1 R/100^3 - 1 - 3R/100]) : (P[1 R/100^2 - 1 - 2R/100]) 29:9

Dividing both sides by P and further simplifying:

[3R/100R/100^2] : [R/100^2] 29:9

This simplifies to:

9(3R/100) 29

Solving for R:

R/100 2/9

Therefore, the required ratio of total interest earned for 3 years and 2 years is:

C3 S3 : C2 S2 P[1 R/100^3 - 1 3R/100] : P[1 R/100^2 - 1 - 2R/100]

Substituting R/100 2/9:

[1 81/100^3 - 1 32/9] : [1 81/100^2 - 1 - 22/9]

This further simplifies to:

[1331/729 - 1/3] : [121/81 - 5/9]

Which finally simplifies to:

272 : 171 [Ans]

Conclusion

In conclusion, understanding the relationship between compound interest (CI) and simple interest (SI) is essential for accurate financial calculations. The example provided in this article demonstrates how to solve complex ratio problems involving these interest calculations. By breaking down the problem into simpler steps, we can find the rate of interest and the ratio of interest earned over different time periods.

For more information on financial calculations, ratios, and other related topics, please continue to explore our resource pages.