Calculating the Annual Interest Rate for Simple Interest

Understanding how to calculate the annual interest rate for simple interest is a fundamental concept in finance and mathematics. This article explores the formula to determine the annual interest rate when the simple interest accumulates to a specific amount over a given period. Let us consider a scenario where we wish to determine the rate at which a sum of money grows to be two-fifths of the principal amount over a period of 10 years.

The Formula and Problem Statement

The formula for simple interest is given by:
SI (P × R × T) / 100

Where:

SI is the simple interest P is the principal amount R is the annual interest rate T is the time in years

The problem requires us to find the annual interest rate (R) when the simple interest is two-fifths (2/5) of the principal (P) after 10 years. Let's break down the solution step by step.

Simplifying the Problem

Given:

SI 2/5 of the principal amount T 10 years

We express the amount (A) as:

[ A P SI ]

Substituting the given SI into the equation:

[ A P (2/5)A ]

Rearranging this equation to isolate A:

[ A - (2/5)A P ]

This simplifies to:

[ (3/5)A P ]

Therefore, expressing A in terms of P:

[ A (5/3)P ]

Substituting and Solving for the Rate (R)

Now, substituting A back into the simple interest formula:

[ SI (2/5)A (2/5)((5/3)P) (2/3)P ]

Recall that: [ SI (P × R × T) / 100 ]

Substituting SI and T:

[ (2/3)P (P × R × 10) / 100 ]

Simplifying further:

[ (2/3) (R × 10) / 100 ]

Multiplying both sides by 100:

[ (200/3) R × 10 ]

Dividing both sides by 10:

[ R (200/30) (20/3) ≈ 6.67 ]

Hence, the annual interest rate R is approximately 6.67%.

Verification and Alternative Approaches

To verify, let's consider a principal amount of Rs 100:

Using Basic Arithmetic

Let the principal be Rs 100, and the rate of interest be r. Given that the simple interest in 10 years is 2/5 of Rs 100, the interest amount is:

[ 100 × (2/5) 40 ]

Using the simple interest formula for 1 year:

[ Interest Principal × Rate × Time ]

With 40 being the interest for 10 years, we can find the rate (r) per annum:

[ 40 100 × r × 10 ]

Thus,

[ r 40 / (1000) 0.04 4% ]

Using Arithmetic Reasoning

If we deduct Rs 10 annually, it will take 6 years to reach Rs 60 (2/5 of Rs 100). This implies an annual interest of Rs 10, so the rate is 4%.

Using Conceptual Understanding

If 3/5 of the principal is 60, then 2/5 of the principal is 40. This annual increment of 40 over 10 years suggests an annual interest rate of 4%.

Conclusion

In conclusion, the annual interest rate that will make the simple interest two-fifths of the principal over a period of 10 years is approximately 6.67%. This can also be verified through simpler methods, such as basic arithmetic and conceptual understanding of interest rates.