Praveena 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 a total of Rs 432 as interest at the end of the loan period, what is the annual rate of interest (in percent)?

Introduction / Context:
This problem checks your understanding of simple interest when the time period is linked directly to the rate of interest. Many exam questions use this pattern, where the number of years is equal to the numerical value of the rate, forcing you to form and solve a quadratic equation in the rate.

Given Data / Assumptions:

• Principal, P = Rs 1,200.
• Rate of interest, R% per annum (unknown).
• Time, T = R years (same numerical value as the rate).
• Total simple interest paid, SI = Rs 432.
• Interest is calculated on simple interest basis only.

Concept / Approach:
The simple interest formula is:
SI = (P * R * T) / 100Here, T is not given directly but is equal to R. Therefore, we substitute T = R, which makes the formula involve R squared. We then solve the resulting quadratic equation to find the correct rate of interest. Since interest rates can not be negative in this context, we select the positive root.

Step-by-Step Solution:
Step 1: Write the simple interest formula with T = R.432 = (1200 * R * R) / 100Step 2: Simplify the constants.432 = 12 * R^2Step 3: Solve for R^2.R^2 = 432 / 12 = 36Step 4: Take the square root of both sides.R = sqrt(36) = 6Step 5: Interpret the result.Therefore, the rate of interest is 6% per annum.

Verification / Alternative check:
We can verify by plugging R = 6 back into the formula. Time T is also 6 years. Then:
SI = (1200 * 6 * 6) / 100 = (1200 * 36) / 100 = 432This exactly matches the given interest amount, confirming that 6% is correct.

Why Other Options Are Wrong:

• 7% would give SI = (1200 * 7 * 7) / 100 = 588, which is greater than Rs 432.
• 9% would give SI = (1200 * 9 * 9) / 100 = 972, much larger than Rs 432.
• 11% would give SI = (1200 * 11 * 11) / 100 = 1,452, which is clearly incorrect.

Common Pitfalls:
Learners often forget to equate the time and the rate or treat R and T as separate variables. Another frequent mistake is to take only the square root step partially or to miscalculate the multiplication. Careful substitution and simplification prevent these errors. Remember that in simple interest, if time equals the numerical rate, the unknown R will often appear squared, so you must be comfortable solving basic quadratic equations that arise from financial formulas.

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