Calculating Ramesh's Investment in a Profit Sharing Partnership

This problem involves the concept of profit sharing in a business partnership. When partners invest different amounts of money for different periods, the profit is distributed in the ratio of the product of their respective investments and the time for which the investment was made. This is often referred to as the (Investment $\times$ Time) ratio.

Understanding the Given Information

• Total Profit = Rs. 9000
• Manoj's Share of Profit = Rs. 6000
• Ramesh's Share of Profit = Total Profit - Manoj's Share = Rs. 9000 - Rs. 6000 = Rs. 3000
• Manoj's Investment = Rs. 20000
• Manoj's Time = 6 months
• Ramesh's Time = 1 year = 12 months
• Let Ramesh's Investment be denoted by 'R'.

Ratio of Profit Shares

The ratio of Manoj's profit share to Ramesh's profit share is:

$\frac{\text{Manoj's Share}}{\text{Ramesh's Share}} = \frac{6000}{3000} = \frac{2}{1}$

So, the ratio of their profit shares is 2:1.

Relating Profit Ratio to Investment-Time Ratio

The profit shares are proportional to the product of investment and time for each partner. Therefore:

$\frac{\text{Manoj's Investment} \times \text{Manoj's Time}}{\text{Ramesh's Investment} \times \text{Ramesh's Time}} = \frac{\text{Manoj's Share}}{\text{Ramesh's Share}}$

Substitute the known values into the equation:

$\frac{20000 \times 6}{R \times 12} = \frac{2}{1}$

Solving for Ramesh's Investment (R)

Now, we need to solve this equation for R, which represents the amount invested by Ramesh.

Simplify the left side of the equation:

$\frac{120000}{12R} = \frac{2}{1}$

Cross-multiply:

$120000 \times 1 = 2 \times 12R$

$120000 = 24R$

Now, isolate R by dividing both sides by 24:

$R = \frac{120000}{24}$

Perform the division:

$R = 5000$

So, the amount invested by Ramesh was Rs. 5000.