Sandeep received Rs. 6000 as his share out of a total profit of Rs. 9000 that he and Saketh earned at the end of one year. If Sandeep invested Rs. 20,000 for 6 months and Saketh kept his investment for the whole year, what was the amount invested by Saketh?

Introduction / Context:
This partnership problem asks you to work backwards from known profit shares and time periods to find an unknown capital investment. It connects ratio of profit, time of investment and the amount of capital each partner contributes to the business.

Given Data / Assumptions:

• Total profit at the end of the year is Rs. 9000.
• Sandeep receives Rs. 6000 from this profit.
• Therefore Saketh receives Rs. 3000.
• Sandeep invested Rs. 20,000 for 6 months.
• Saketh invested some amount for the full 12 months.
• We must find the amount invested by Saketh.

Concept / Approach:
In partnerships, profit sharing is proportional to the product of capital and time. If one partner invests capital A1 for T1 months and another invests capital A2 for T2 months, then their profit shares are in the ratio A1 * T1 : A2 * T2. Here we know the profit ratio and one capital value with time, so we can find the unknown capital by forming a proportion and solving for the missing quantity.

Step-by-Step Solution:
Step 1: Determine the ratio of profit shares. Sandeep's share = Rs. 6000 and Saketh's share = Rs. 3000. Step 2: Profit ratio Sandeep : Saketh = 6000 : 3000 = 2 : 1. Step 3: Compute Sandeep's capital time product: 20,000 * 6 = 1,20,000. Step 4: Let Saketh's investment be x rupees for the full 12 months. Step 5: His capital time product is x * 12 = 12x. Step 6: Set up the proportion using capital time products: 1,20,000 : 12x = 2 : 1. Step 7: Cross multiply: 1,20,000 * 1 = 2 * 12x, which gives 1,20,000 = 24x. Step 8: Solve for x: x = 1,20,000 / 24 = 5000. Step 9: Therefore, Saketh invested Rs. 5000 for the whole year.

Verification / Alternative check:
If Saketh invested Rs. 5000 for 12 months, his capital time product is 5000 * 12 = 60,000. Sandeep's capital time product is 20,000 * 6 = 1,20,000. The ratio of their capital time products is 1,20,000 : 60,000 = 2 : 1, which matches the ratio of their profit shares 6000 : 3000. This confirms that the investment of Rs. 5000 by Saketh is correct.

Why Other Options Are Wrong:
Rs. 6200, Rs. 7600 and Rs. 8100 would produce capital time ratios that do not simplify to 2 : 1 when compared with Sandeep's product of 1,20,000. Therefore, none of these alternatives are consistent with the given profit distribution and time periods.

Common Pitfalls:
A common mistake is to compare only the capital amounts and ignore the time for which each amount is invested. Another error is to assume that larger capital always gets larger profit without checking the proportional relationship with time. Always multiply capital by time first to get a fair comparison in partnership questions.

Final Answer:
The amount invested by Saketh is Rs. 5000.