Maneela lends a total of Rs. 8000, part at 5% per annum and the rest at 6% per annum simple interest. If the total interest she earns in 2 years is Rs. 820, in what ratio should the Rs. 8000 be divided between the two rates?

Introduction / Context:
This question checks understanding of simple interest and allocation of a total investment between two different interest rates. The aim is to determine how much money is placed at each rate when the overall interest earned over a fixed time period is known. It is a typical mixture-of-rates question based on simple interest.

Given Data / Assumptions:
• Total amount lent by Maneela = Rs. 8000.
• Part of the amount is lent at 5% per annum simple interest.
• The remaining part is lent at 6% per annum simple interest.
• Time period for both investments = 2 years.
• Total interest received after 2 years = Rs. 820.

Concept / Approach:
Let the amount lent at 5% be x rupees, and the amount lent at 6% be 8000 − x rupees. For simple interest, interest = Principal * Rate * Time / 100. We form an equation using the total interest from both parts and set it equal to Rs. 820. Solving this equation gives the individual principal values and thus the ratio between them. This is a direct algebraic application of the simple interest formula.

Step-by-Step Solution:
Step 1: Let the amount lent at 5% be x and at 6% be (8000 − x).Step 2: Interest from x at 5% for 2 years = x * 5 * 2 / 100 = 0.10x.Step 3: Interest from (8000 − x) at 6% for 2 years = (8000 − x) * 6 * 2 / 100 = 0.12(8000 − x).Step 4: Total interest is given as 820, so 0.10x + 0.12(8000 − x) = 820.Step 5: Simplify: 0.10x + 960 − 0.12x = 820 ⇒ −0.02x + 960 = 820 ⇒ −0.02x = −140 ⇒ x = 7000.Step 6: Amount at 5% = 7000 and amount at 6% = 8000 − 7000 = 1000.Step 7: Required ratio (5% : 6%) = 7000 : 1000 = 7 : 1.

Verification / Alternative check:
Compute the actual interest using the found amounts. From Rs. 7000 at 5% for 2 years, interest = 7000 * 5 * 2 / 100 = Rs. 700. From Rs. 1000 at 6% for 2 years, interest = 1000 * 6 * 2 / 100 = Rs. 120. Total interest = 700 + 120 = Rs. 820, which matches the given value. This confirms that the ratio 7 : 1 is correct.

Why Other Options Are Wrong:
Ratio 13 : 5 would correspond to very different amounts and would not produce a total interest of Rs. 820 over 2 years at the given rates.
Ratios 15 : 7 and 2 : 7 also lead to principal allocations that, when plugged into the simple interest formula, give total interest either more or less than Rs. 820. Thus they conflict with the problem data.

Common Pitfalls:
One common mistake is to treat 5% and 6% as if they can be averaged directly without accounting for the different amounts invested at each rate. Another error is to forget that the time period is the same for both parts and to incorrectly use different times. Setting up the algebraic equation correctly from the total interest is the most reliable method.

Final Answer:
The Rs. 8000 should be divided in the ratio 7 : 1 between 5% and 6% simple interest.