Reena takes a loan of Rs 1,200 at simple interest for as many years as the numerical value of the rate of interest; if she pays Rs 432 as interest at the end of the loan period, what is the annual rate of simple interest in percent?

Introduction / Context:
This question is almost identical in structure to an earlier problem where the number of years equals the numerical value of the rate of interest. Reena borrows money at simple interest with this special relationship between time and rate, and the total interest paid is known. The goal is to deduce the annual rate, which again involves forming and solving a quadratic relation in R based on the simple interest formula.

Given Data / Assumptions:
- Principal borrowed P = Rs 1,200. - Total simple interest paid SI = Rs 432. - Let the annual simple interest rate be R percent. - Time period T in years is equal to R, the numerical value of the rate. - Interest is calculated strictly using the simple interest formula.

Concept / Approach:
The simple interest formula is SI = (P * R * T) / 100. The condition T = R leads to SI = (P * R^2) / 100. Substituting P and SI yields an equation in R^2, which we can solve to get R. Because R represents an interest rate, we only consider the positive root. This algebraic step is crucial for correctly interpreting such special condition problems in simple interest.

Step-by-Step Solution:
Step 1: Start from SI = (P * R * T) / 100. Step 2: Use T = R to get SI = (P * R^2) / 100. Step 3: Substitute SI = 432 and P = 1,200. Step 4: Write the equation: 432 = (1,200 * R^2) / 100. Step 5: Multiply both sides by 100: 432 * 100 = 1,200 * R^2. Step 6: 432 * 100 = 43,200, so 43,200 = 1,200 * R^2. Step 7: Divide both sides by 1,200: R^2 = 43,200 / 1,200 = 36. Step 8: Take the positive square root: R = square root of 36 = 6. Step 9: Therefore, the annual rate of simple interest is 6 percent per annum.

Verification / Alternative check:
Verify by substituting R = 6 percent and T = 6 years back into the formula. SI = (1,200 * 6 * 6) / 100. First 1,200 * 6 = 7,200, then 7,200 * 6 = 43,200, and dividing by 100 gives SI = Rs 432. This matches the given interest amount exactly, confirming that a 6 percent rate is correct.

Why Other Options Are Wrong:
A rate of 3.6 percent would require R^2 to be 12.96, which does not satisfy the equation 1,200 * R^2 / 100 = 432. A rate of 18 percent gives R^2 = 324, resulting in an interest much larger than Rs 432. The option stating that the rate cannot be determined is incorrect because we obtained a unique positive solution. Only 6 percent satisfies the given conditions.

Common Pitfalls:
Learners sometimes forget that T equals R, and instead treat T as an independent value. Others may incorrectly handle the percent conversion or miscalculate R^2. Recognising that the condition T = R transforms the simple interest formula into a quadratic in R is the key to solving this smoothly.

Final Answer:
The annual rate of simple interest for Reena's loan is 6% per annum.