A sum of Rs. 50,000 is divided into two parts. One part is lent at 10 percent simple interest per annum and the other part is lent at 20 percent simple interest per annum. At the end of one year, the total interest received is Rs. 7,000. How much money was lent at 10 percent?

Introduction / Context:
This is a straightforward simple interest and ratio type question. The total principal is split into two parts, each earning a different rate of simple interest. We are given the total interest earned after one year and must determine how much of the original sum was lent at the lower interest rate of 10 percent.

Given Data / Assumptions:

• Total principal = Rs. 50,000.
• First part is lent at 10 percent simple interest per annum.
• Second part is lent at 20 percent simple interest per annum.
• Total interest received after 1 year = Rs. 7,000.
• Time period for both loans is the same, namely 1 year.

Concept / Approach:
Simple interest for one year is calculated as Principal * Rate * Time, where time is measured in years and rate is in decimal form. Because time is one year, interest is directly proportional to the principal and the rate. Let one part of the principal be x and the other part be (50,000 - x). We express the total interest as the sum of the interest on both parts and equate that to Rs. 7,000, then solve for x.

Step-by-Step Solution:
Step 1: Let the amount lent at 10 percent be x rupees.Step 2: Then the amount lent at 20 percent is 50,000 - x rupees.Step 3: Simple interest on x at 10 percent for 1 year = 0.10 * x = 0.1x.Step 4: Simple interest on (50,000 - x) at 20 percent for 1 year = 0.20 * (50,000 - x) = 10,000 - 0.2x.Step 5: Total interest after 1 year = 0.1x + (10,000 - 0.2x) = 10,000 - 0.1x.Step 6: The total interest is given as Rs. 7,000, so set 10,000 - 0.1x = 7,000.Step 7: Rearranging, -0.1x = 7,000 - 10,000 = -3,000.Step 8: Hence x = 3,000 / 0.1 = Rs. 30,000.

Verification / Alternative check:
If Rs. 30,000 is lent at 10 percent, it earns 3,000 interest in one year. The remaining amount is 50,000 - 30,000 = Rs. 20,000, lent at 20 percent, which earns 4,000 interest in one year. Total interest is 3,000 + 4,000 = Rs. 7,000, which exactly matches the given data. This verification confirms the correctness of the answer.

Why Other Options Are Wrong:
If Rs. 20,000 were lent at 10 percent, interest would be 2,000, and the remaining 30,000 at 20 percent would produce 6,000, giving a total of 8,000, not 7,000. If Rs. 10,000 were lent at 10 percent, total interest would be far smaller. If Rs. 40,000 were lent at 10 percent, then 10 percent on 40,000 is 4,000 and 20 percent on 10,000 is 2,000, giving only 6,000. Thus these values are inconsistent with the given interest of 7,000. 'None of these' is incorrect because Rs. 30,000 fits perfectly.

Common Pitfalls:
Some students mistakenly divide 7,000 in proportion to the interest rates without considering the constraint that the total principal is 50,000. Others may erroneously assume equal division. Always set up an equation based on the exact definition of simple interest and the given total interest.

Final Answer:
The amount of money lent at 10 percent interest is Rs. 30,000.