A sum of Rs. 2600 is lent out in two parts: one part at 9% simple interest per annum and the other part at 7% simple interest per annum.\nIf the total annual interest from both parts together is Rs. 206, how much money was lent at 7%?

Introduction / Context:
This is a classic simple interest and linear equation problem. A fixed total amount is split between two different interest rates, and the total annual interest is known. You are asked to find how much of the principal was invested at the lower rate. Such questions test your ability to form and solve linear equations involving percentages.

Given Data / Assumptions:

• Total principal lent = Rs. 2600.
• Part of the money is lent at 9% simple interest per annum.
• The remaining part is lent at 7% simple interest per annum.
• Total annual interest (from both parts together) = Rs. 206.
• We must find the amount lent at 7%.
• Interest is simple and calculated for one year.

Concept / Approach:
Let the amount lent at 9% be x. Then the amount lent at 7% is 2600 − x. The annual simple interest from a principal P at rate R% is (P * R / 100). Using this, we write expressions for interest from each part and add them to match the given total interest of Rs. 206. This gives a linear equation in x, which can be solved by basic algebra. After finding x, we subtract it from 2600 to get the amount at 7%.

Step-by-Step Solution:
Step 1: Let the amount lent at 9% be x rupees. Step 2: Then the amount lent at 7% is 2600 − x rupees. Step 3: Annual interest from the 9% part = x * 9 / 100 = 0.09x. Step 4: Annual interest from the 7% part = (2600 − x) * 7 / 100 = 0.07(2600 − x). Step 5: Total annual interest = 0.09x + 0.07(2600 − x) = 206. Step 6: Expand: 0.09x + 0.07 * 2600 − 0.07x = 206. Step 7: Compute 0.07 * 2600 = 182. Step 8: Combine like terms: (0.09x − 0.07x) + 182 = 206 ⇒ 0.02x + 182 = 206. Step 9: Subtract 182 from both sides: 0.02x = 24. Step 10: Solve for x: x = 24 / 0.02 = 1200. Step 11: Amount lent at 7% = 2600 − 1200 = 1400.

Verification / Alternative check:
Check interest values for Rs. 1200 at 9% and Rs. 1400 at 7%. Interest from 1200 at 9% = 1200 * 9 / 100 = 108. Interest from 1400 at 7% = 1400 * 7 / 100 = 98. Total interest = 108 + 98 = 206, which matches the given total. Thus the split is verified, and Rs. 1400 is correctly identified as the amount lent at 7%.

Why Other Options Are Wrong:
Rs. 900, Rs. 1600, Rs. 1200, Rs. 1000: Substituting any of these values as the 7% part leads to total interest values different from Rs. 206. For example, if 1600 were lent at 7%, the remaining 1000 at 9% would produce 112 + 112 = 224 in interest, not 206.

Common Pitfalls:
Students sometimes mistakenly assign x to the 7% part instead of the 9% part and then misinterpret the final result. Another common error is in handling the signs when expanding 0.07(2600 − x), especially forgetting the negative sign in front of 0.07x. Carefully forming the equation and checking each expansion step helps avoid these mistakes.

Final Answer:
The amount of money lent at 7% is Rs. 1400.