At what annual rate of simple interest will a principal of Rs 415 grow to an amount of Rs 514 in 4 years, assuming interest is calculated using the simple interest formula only?

Introduction / Context:
This question asks for the annual simple interest rate that causes a principal to grow to a given amount over a fixed time. It tests the ability to extract the interest earned from the final amount and then use the simple interest formula to find the rate of interest. Problems of this type are very common in aptitude tests and financial arithmetic.

Given Data / Assumptions:
- Principal P = Rs 415. - Final amount A after 4 years = Rs 514. - Time period T = 4 years. - Interest is paid using simple interest, not compound interest. - Annual rate of simple interest R percent per annum is to be found.

Concept / Approach:
In simple interest, the amount A after T years is given by A = P + SI, where SI is simple interest. First we compute SI as A minus P. Then we use the formula SI = (P * R * T) / 100 and rearrange to get R = (SI * 100) / (P * T). Substituting the known values for SI, P, and T yields the required rate in percent per annum.

Step-by-Step Solution:
Step 1: Compute the interest earned: SI = A − P = 514 − 415. Step 2: Calculate 514 − 415 = 99, so SI = Rs 99. Step 3: Use the simple interest formula SI = (P * R * T) / 100. Step 4: Rearrange for R: R = (SI * 100) / (P * T). Step 5: Substitute SI = 99, P = 415, T = 4. Step 6: Compute numerator: 99 * 100 = 9,900. Step 7: Compute denominator: 415 * 4 = 1,660. Step 8: Divide to get R: R = 9,900 / 1,660 which is approximately 5.963855 percent. Step 9: Rounding to two decimal places, R is approximately 5.96 percent per annum.

Verification / Alternative check:
To verify, assume R = 5.96 percent and recompute the simple interest. SI = (415 * 5.96 * 4) / 100. First, 415 * 5.96 ≈ 2,474. Then 2,474 * 4 ≈ 9,896 and dividing by 100 gives about Rs 98.96, which is extremely close to Rs 99, the required interest. The small difference is due to rounding; hence 5.96 percent is the best match among the given options.

Why Other Options Are Wrong:
A rate of 4.58 percent gives SI = (415 * 4.58 * 4) / 100, which is much less than Rs 99. A rate of 6.52 percent gives SI higher than Rs 99. Similarly, 4.98 percent also produces interest significantly different from 99 when substituted into the formula. Only 5.96 percent generates an interest amount that matches the difference between 514 and 415.

Common Pitfalls:
Candidates sometimes forget to subtract the principal from the final amount and mistakenly treat A as SI. Another error is to use time in months rather than years or to round too early in the calculation, leading to inaccurate results. Keeping all intermediate values in exact fractional or decimal form until the final step yields a more accurate rate.

Final Answer:
The required annual rate of simple interest is approximately 5.96% per annum.