Understanding the Business Partnership Problem

This problem involves calculating the capital contribution of one partner (M) in a business venture with another partner (N). We are given information about their investments, the duration each invested, the difference in their profits, and the total profit earned. The goal is to find the specific amount M invested.

Key Information Given:

• M invests ₹68,000 more than N.
• M invested for 4 months.
• N invested for 7 months.
• M's profit share is ₹856 more than N's profit share.
• The total profit is ₹6,848.

Step 1: Calculate Individual Profit Shares

We know the total profit and the difference between M's and N's profit. We can set up two equations to find their individual profit shares:

Let $M_{profit}$ be M's profit share and $N_{profit}$ be N's profit share.

1. $M_{profit} + N_{profit} = 6848$ (Total Profit)
2. $M_{profit} = N_{profit} + 856$ (M's profit is ₹856 more than N's)

Substitute the second equation into the first:

$(N_{profit} + 856) + N_{profit} = 6848$

$2 \times N_{profit} + 856 = 6848$

$2 \times N_{profit} = 6848 - 856$

$2 \times N_{profit} = 5992$

$N_{profit} = \frac{5992}{2} = 2996$

So, N's profit share is ₹2,996.

Now, find M's profit share using the second equation:

$M_{profit} = N_{profit} + 856 = 2996 + 856 = 3852$

M's profit share is ₹3,852.

Step 2: Determine the Ratio of Profit Distribution

In a partnership, profit is distributed based on the investment made and the duration for which the investment was made. The ratio of profit sharing is equal to the ratio of the product of capital and time for each partner.

Let $M_{invest}$ be M's capital and $N_{invest}$ be N's capital.

M's investment duration ($M_{time}$) = 4 months.

N's investment duration ($N_{time}$) = 7 months.

The ratio of profit sharing is:

$ \frac{M_{profit}}{N_{profit}} = \frac{M_{invest} \times M_{time}}{N_{invest} \times N_{time}} $

Substitute the calculated profit shares and given durations:

$ \frac{3852}{2996} = \frac{M_{invest} \times 4}{N_{invest} \times 7} $

Step 3: Simplify the Ratio and Set Up the Equation for Capital

First, simplify the profit ratio $\frac{3852}{2996}$. Both numbers are divisible by 4:

$ \frac{3852 \div 4}{2996 \div 4} = \frac{963}{749} $

We can further check for common factors. Both 963 and 749 are divisible by 107:

$ \frac{963 \div 107}{749 \div 107} = \frac{9}{7} $

So, the profit ratio is $\frac{9}{7}$.

Now, let's express the capital relationship. Let N's capital be $x$. Then M's capital is $x + 68000$.

Substituting these into the ratio equation:

$ \frac{9}{7} = \frac{(x + 68000) \times 4}{x \times 7} $

Step 4: Solve for N's Capital (x)

We can simplify the equation by multiplying both sides by 7:

$ \frac{9}{7} \times 7 = \frac{(x + 68000) \times 4}{x \times 7} \times 7 $

$ 9 = \frac{(x + 68000) \times 4}{x} $

Multiply both sides by $x$:

$ 9x = (x + 68000) \times 4 $

Distribute the 4 on the right side:

$ 9x = 4x + (68000 \times 4) $

$ 9x = 4x + 272000 $

Subtract $4x$ from both sides:

$ 9x - 4x = 272000 $

$ 5x = 272000 $

Solve for $x$:

$ x = \frac{272000}{5} $

$ x = 54400 $

So, N's capital contribution is ₹54,400.

Step 5: Calculate M's Capital Contribution

We know that M invested ₹68,000 more than N.

M's Capital = N's Capital + ₹68,000

M's Capital = $x + 68000$

M's Capital = $54400 + 68000$

M's Capital = $122400$

Conclusion

The capital contributed by M is ₹1,22,400.