Arun took a loan of Rs 29,000 at simple interest for as many years as the numerical value of the rate of interest. If he paid Rs 10,440 as interest at the end of the loan period, what was the rate of interest per annum under this simple interest arrangement?

Introduction / Context:
This question is a classic aptitude problem where the rate of interest and the time period in years are numerically equal. It uses the simple interest formula to form a quadratic-like relation in the rate and then solves for that rate. Understanding how to translate verbal conditions such as "for as many years as the rate" into algebra is important in many similar questions.

Given Data / Assumptions:

Principal P = Rs 29,000.
Rate of interest per annum = r%.
Time T in years is numerically equal to r (so T = r years).
Simple interest paid at the end of the period = Rs 10,440.
Formula: SI = (P * R * T) / 100.

Concept / Approach:
Since T = r, the simple interest formula becomes SI = (P * r * r) / 100. That is, SI depends on r squared. We substitute the known values of P and SI into this relation and solve for r. Because r represents a rate of interest, we look for a positive value that matches one of the given options.

Step-by-Step Solution:
Step 1: Use SI = (P * r * T) / 100 with T = r and P = 29000. Step 2: Then SI = (29000 * r * r) / 100 = (29000 * r^2) / 100. Step 3: Given SI = 10440, set up the equation 10440 = (29000 * r^2) / 100. Step 4: Multiply both sides by 100 to remove the denominator: 10440 * 100 = 29000 * r^2. Step 5: So, 1044000 = 29000 * r^2. Step 6: Divide both sides by 29000: r^2 = 1044000 / 29000 = 36. Step 7: Take the positive square root since rate cannot be negative: r = 6. Step 8: Therefore, the rate of interest per annum is 6%.

Verification / Alternative check:
Check by computing the interest directly. If r = 6%, then T = 6 years. SI = (29000 * 6 * 6) / 100 = (29000 * 36) / 100 = 1044000 / 100 = 10440. This matches the given simple interest, confirming that the rate must be 6% per annum.

Why Other Options Are Wrong:
If r = 5.5%, the interest becomes (29000 * 5.5 * 5.5) / 100, which is not 10440. Similarly, substituting 6.5%, 7%, or 4.5% into the formula produces interest amounts different from Rs 10,440. Thus, none of these alternative rates satisfy the equation, so they are incorrect.

Common Pitfalls:
Students sometimes mistakenly set T equal to r% instead of r years, or they forget to square r when T = r. Another error is failing to multiply the given simple interest by 100 while clearing the denominator. Careful algebra and a quick verification step help avoid such mistakes.

Final Answer:
The correct rate of interest per annum is 6%.