The difference between the simple interest earned when Rs P is invested for 4 years at 9% per annum and when the same sum Rs P is invested for 2 years at 12% per annum is Rs 480. What is the value of P?

Introduction / Context:
This question compares two different simple interest scenarios for the same principal but with different rates and time periods. The difference between the two interest amounts is given, and the task is to find the principal P. Such problems demonstrate how simple interest scales directly with both rate and time and are frequently used to test algebraic manipulation with percentages in aptitude exams.

Given Data / Assumptions:

Principal in both cases = Rs P
Case 1: Rate = 9 percent per annum, time = 4 years
Case 2: Rate = 12 percent per annum, time = 2 years
Difference between interest in Case 1 and Case 2 = Rs 480
Both cases use simple interest

Concept / Approach:
Use the simple interest formula I = P * r * t / 100 for each case. Express I1 and I2 in terms of P, then write an equation representing the difference I1 minus I2 equal to 480 rupees. Solving this linear equation yields the value of P. Because the same principal is used in both cases, P appears as a common factor and the equation simplifies nicely.

Step-by-Step Solution:
Step 1: Express interest in Case 1. I1 = P * 9 * 4 / 100 = P * 36 / 100 = 0.36 * P. Step 2: Express interest in Case 2. I2 = P * 12 * 2 / 100 = P * 24 / 100 = 0.24 * P. Step 3: The difference between the two interests is given. I1 - I2 = 480. Step 4: Substitute the expressions in terms of P. 0.36 * P - 0.24 * P = 480. Step 5: Simplify the left side. (0.36 - 0.24) * P = 0.12 * P. Step 6: So 0.12 * P = 480. Step 7: Solve for P: P = 480 / 0.12. Step 8: P = 4,000 rupees.

Verification / Alternative check:
Check by substituting P = 4,000. Case 1 interest at 9 percent for 4 years: I1 = 4,000 * 9 * 4 / 100 = 4,000 * 36 / 100 = 1,440 rupees. Case 2 interest at 12 percent for 2 years: I2 = 4,000 * 12 * 2 / 100 = 4,000 * 24 / 100 = 960 rupees. Difference I1 - I2 = 1,440 - 960 = 480 rupees, which matches the given difference and confirms that P is 4,000 rupees.

Why Other Options Are Wrong:
If P were Rs 2,000, then 0.12 * P would be only 240 rupees, not 480 rupees. For P = 2,500, 0.12 * P = 300 rupees. For P = 3,500, 0.12 * P = 420 rupees. None of these match the given difference. Only P = 4,000 rupees produces the required difference of 480 rupees between the two simple interest amounts.

Common Pitfalls:
Some learners mistakenly subtract rates or times directly without forming proper interest expressions. Others may reverse the difference and set I2 - I1 equal to 480, which leads to a negative coefficient for P. Additionally, incorrect handling of percentages, such as forgetting to divide by 100, can distort results. Writing out the formula clearly for each case and then carefully subtracting the expressions helps to prevent such errors.

Final Answer:
The principal P invested in both cases is Rs 4,000.