A person divides a sum of Rs. 17,200 into three parts and invests them at simple interest rates of 5%, 6% and 9% per annum respectively. At the end of 2 years, the simple interest earned on each part is the same. What is the amount (in rupees) invested at 9% per annum?

Introduction / Context:
This problem extends the idea of splitting a total sum among investments with different simple interest rates, such that each part earns the same interest over a fixed time. It is a classic aptitude question where you must use proportional reasoning and the simple interest formula. The time is the same for all three parts, and only the rates differ. Because the interests are equal, the amounts invested must be in inverse proportion to the rates of interest. The goal is to determine how much was invested at the highest rate of 9% per annum.

Given Data / Assumptions:

• Total sum = Rs. 17,200.
• Three simple interest rates: 5%, 6%, and 9% per annum.
• Time T = 2 years for all three investments.
• Simple interest earned on each part after 2 years is the same.
• All investments follow the standard simple interest formula with constant rates.

Concept / Approach:
For simple interest, SI = P * R * T / 100. When T is the same for all parts and equal interest is earned, we have P1 * R1 = P2 * R2 = P3 * R3 (up to a constant factor). This means the principal amounts are inversely proportional to the product R * T, but since T is the same for all, they are inversely proportional to the rates alone. Thus, the ratio of the three parts is 1/5 : 1/6 : 1/9. After simplifying this ratio and using the total sum, we can find the actual amount corresponding to the part invested at 9%.

Step-by-Step Solution:
Step 1: Let the amounts invested at 5%, 6%, and 9% be A, B, and C respectively. Step 2: Because T = 2 years for each, SI for each is proportional to P * R. Step 3: Equal interest implies A * 5 = B * 6 = C * 9. Step 4: Therefore, A : B : C is inversely proportional to 5 : 6 : 9. Step 5: So A : B : C = 1/5 : 1/6 : 1/9. Step 6: To remove fractions, take the LCM of 5, 6, and 9, which is 90, and multiply: 1/5 : 1/6 : 1/9 = 18 : 15 : 10. Step 7: Thus A : B : C = 18 : 15 : 10. Step 8: Sum of ratio parts = 18 + 15 + 10 = 43. Step 9: Total amount is Rs. 17,200, so each part = 17200 / 43 = 400 rupees. Step 10: Amount at 9% = C = 10 parts = 10 * 400 = Rs. 4000.

Verification / Alternative check:
Check the interests over 2 years. At 5%: A = 18 * 400 = 7200, SI = 7200 * 5 * 2 / 100 = 720. At 6%: B = 6000, SI = 6000 * 6 * 2 / 100 = 720. At 9%: C = 4000, SI = 4000 * 9 * 2 / 100 = 720. The interest from each part is Rs. 720, and the total capital is 7200 + 6000 + 4000 = 17,200, which matches the given data. This confirms that Rs. 4000 is correctly invested at 9%.

Why Other Options Are Wrong:
If 3200 were at 9%, interest would be 3200 * 9 * 2 / 100 = 576, which cannot match the interest from the other parts under the given total. Similarly, 4800 or 5000 at 9% would give interests of 864 and 900 respectively, which do not align with equal interest across three parts for a total of Rs. 17,200. Only Rs. 4000 gives a consistent solution with equal interest on each portion.

Common Pitfalls:
Students often wrongly assume the amounts are proportional to the rates directly instead of inversely. Another error is forgetting that the time is the same for all three parts and mistakenly including T in the ratio. Some learners attempt to distribute the total sum by guesswork instead of systematically using ratios, which can be very inefficient and error-prone. Using inverse proportion is the key to solving these kinds of problems quickly and accurately.

Final Answer:
The amount invested at 9% per annum is Rs. 4000.