A sum of Rs. 15 200 is divided into two parts. The simple interest on the first part at 25% per annum is equal to the simple interest on the second part at 13% per annum for the same time. What is the simple interest (in rupees) on each part for 1 year?

Introduction / Context:
This question involves splitting a total principal into two parts that earn simple interest at different rates. The condition is that the interest from both parts is equal for the same time period. You are asked to find the amount of interest on each part for 1 year.

Given Data / Assumptions:

• Total principal = Rs. 15 200.
• First part of principal = P1, interest rate R1 = 25% per annum.
• Second part of principal = P2, interest rate R2 = 13% per annum.
• P1 + P2 = 15 200.
• Simple interest on P1 at 25% equals simple interest on P2 at 13% for the same time.
• Time is taken as 1 year for finding the interest in rupees on each part.

Concept / Approach:
Simple interest for 1 year is SI = (P * R * 1) / 100 = P * R / 100. The condition that interests are equal gives (P1 * 25) / 100 = (P2 * 13) / 100. That leads to a relation between P1 and P2. Combine this with P1 + P2 = 15 200 to solve for both principals, then compute the interest amount on either part for 1 year.

Step-by-Step Solution:
Step 1: From equality of interests: (P1 * 25) / 100 = (P2 * 13) / 100. Step 2: Multiply both sides by 100 to remove denominators: 25 * P1 = 13 * P2. Step 3: Express P2 in terms of P1: P2 = (25 * P1) / 13. Step 4: Use total principal: P1 + P2 = 15 200. Step 5: Substitute P2: P1 + (25 * P1) / 13 = 15 200. Step 6: Combine terms: (13 * P1 + 25 * P1) / 13 = (38 * P1) / 13 = 15 200. Step 7: Solve for P1: P1 = 15 200 * 13 / 38 = 5200. Step 8: Then P2 = 15 200 - 5200 = 10 000. Step 9: Interest on P1 in 1 year: SI1 = (5200 * 25) / 100 = 1300. Step 10: Interest on P2 in 1 year: SI2 = (10 000 * 13) / 100 = 1300, confirming equality.

Verification / Alternative check:
Both parts generate Rs. 1300 as interest in 1 year, satisfying the condition that the interests are equal. The total interest for 1 year is 1300 + 1300 = 2600, and the distribution of principal P1 = 5200 and P2 = 10 000 is consistent with the total of Rs. 15 200.

Why Other Options Are Wrong:
2500, 3250, and 1625 are not equal to the actual equal interest of Rs. 1300 on each part. They would either break the equality condition or not match the simple interest formula for the solved principals.

Common Pitfalls:
Some students mistakenly equate P1 and P2 or incorrectly set up the interest equality. Others forget to use the total principal constraint. Correctly forming and solving the two equations is essential.

Final Answer:
The simple interest of each part for 1 year is 1300 rupees.