A total sum of Rs. 10,000 is lent out partly at 6% per annum and the remaining amount at 10% per annum, both on simple interest. If the average rate of simple interest earned on the whole sum is 9.2% per annum, how much money, in rupees, is lent at 10% per annum?

Introduction / Context:
This is a classic mixture or allocation question involving different simple interest rates. The total principal is divided into two parts lent at 6% and 10% per annum, and we are told the effective average rate on the entire Rs. 10,000 is 9.2%. Using this average, we determine how much was invested at the higher rate of 10% per annum.

Given Data / Assumptions:

• Total principal P_total = Rs. 10,000.
• Let x be the amount lent at 10% per annum.
• Then (10000 - x) is lent at 6% per annum.
• Average simple interest rate on entire sum = 9.2% per annum.
• Same time period is assumed for both parts, so time cancels out in average rate calculations.

Concept / Approach:
Because the time is the same for both parts, the overall average rate depends only on the weighted average of the two rates, using principal amounts as weights. We equate the total interest per year from both parts to 9.2% of the full Rs. 10,000 and solve for x, the amount at 10%.

Step-by-Step Solution:
Interest per year from the amount at 10%: I1 = x * 10 / 100 = 0.10x. Interest per year from the amount at 6%: I2 = (10000 - x) * 6 / 100 = 0.06 * (10000 - x). Total yearly interest = I1 + I2. Average yearly interest at 9.2% on Rs. 10000 = 10000 * 9.2 / 100 = Rs. 920. Form equation: 0.10x + 0.06 * (10000 - x) = 920. Simplify: 0.10x + 600 - 0.06x = 920. Combine x terms: 0.04x + 600 = 920. So 0.04x = 320, giving x = 320 / 0.04 = 8000.
Verification / Alternative check:
Check interest: At 10%, Rs. 8000 earns 800 per year. At 6%, Rs. 2000 earns 120 per year. Total interest per year = 800 + 120 = 920. On Rs. 10000, 920 corresponds to an average rate of 9.2%, confirming that Rs. 8000 is correct.

Why Other Options Are Wrong:
Rs. 5000, Rs. 6500, Rs. 8500, and Rs. 9000 all yield total interest per year that is not equal to Rs. 920. Each incorrect allocation either gives a lower or higher average rate than 9.2% when the total interest is divided by 10000, so they do not satisfy the condition.

Common Pitfalls:
Some students try to apply a simple average of 6% and 10% without considering the different principal portions, which would give 8%, not 9.2%. Others set up the equation incorrectly by mixing up which amount is at which rate. Remember that average rate must be based on weighted contributions of each part.

Final Answer:
The amount lent at 10% per annum is Rs. 8000.