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 interest?

Introduction / Context:
This question is structurally the same type as other problems where the number of years is equal to the numerical value of the rate of interest. It is designed to test whether you can correctly set up and solve a quadratic equation in the rate when only the total simple interest and principal are directly given.

Given Data / Assumptions:

• Principal, P = Rs 1,200.
• Rate of interest, R% per annum (unknown).
• Time of loan, T = R years (numerical value equal to the rate).
• Total simple interest paid, SI = Rs 432.

Concept / Approach:
The simple interest formula is:
SI = (P * R * T) / 100Here T is not an independent variable but is equal to R, so T = R. Substituting T = R into the formula will give a quadratic relation in R. Solving this quadratic will yield the required interest rate. Since rates are taken as positive, we choose the positive root.

Step-by-Step Solution:
Step 1: Substitute T = R into the simple interest formula.432 = (1200 * R * R) / 100Step 2: Simplify the expression.432 = 12 * R^2Step 3: Solve for R^2.R^2 = 432 / 12 = 36Step 4: Take the square root.R = sqrt(36) = 6Hence, R = 6% per annum.

Verification / Alternative check:
Verify by substituting R = 6 back into the formula. Time T is also 6 years. Then:
SI = (1200 * 6 * 6) / 100= (1200 * 36) / 100= 43,200 / 100= Rs 432The computed simple interest equals the given interest, confirming that the rate of interest is indeed 6% per annum.

Why Other Options Are Wrong:

• 3.6% would give much lower simple interest when used in the formula and would not produce Rs 432.
• 18% would give a much higher simple interest, far exceeding Rs 432.
• Cannot be determined is incorrect because the data provided is sufficient to form and solve the quadratic equation in R.

Common Pitfalls:
Students sometimes treat time and rate as separate and try to assume arbitrary values, which does not work here. Others might forget to divide by 100 correctly or mishandle the square root, leading to an incorrect rate. It is also important to remember that only the positive root is meaningful in the context of interest rates. Writing each algebraic step clearly ensures that the correct rate is obtained.

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