Solving the Business Partnership Problem

This solution explains how to find the capital contributed by partner M in a business venture, considering different investment durations and profit shares.

Understanding the Problem Details

We are given a business scenario involving two partners, M and N. Key information includes:

• M invests ₹79,000 more than N.
• M invests for 7 months, and N invests for 6 months.
• M's profit share is ₹745 more than N's profit share.
• The total profit of the business is ₹6,705.
• The objective is to find the capital invested by M.

Calculating Individual Profit Shares

The total profit is divided between M and N. We know the total profit and the difference between their shares. Let's denote M's profit share as P_M and N's profit share as P_N.

• Total Profit: P_M + P_N = ₹6,705
• Profit Difference: P_M = P_N + ₹745

To find the individual shares, we substitute the second equation into the first:

(P_N + ₹745) + P_N = ₹6,705

Combine like terms:

2P_N + ₹745 = ₹6,705

Now, we isolate P_N:

2P_N = ₹6,705 - ₹745

2P_N = ₹5,960

Solve for P_N:

P_N = ₹5,960 / 2

P_N = ₹2,980

Using the profit difference, we find M's profit share:

P_M = P_N + ₹745

P_M = ₹2,980 + ₹745

P_M = ₹3,725

So, M's profit share is ₹3,725 and N's profit share is ₹2,980.

Calculating Capital Contributions

Profit distribution in partnerships is proportional to the product of capital invested and the duration of investment. Let C_M represent the capital invested by M and C_N represent the capital invested by N.

The ratio of profits is related to capital and time as follows:

P_M : P_N = (C_M × Time_M) : (C_N × Time_N)

Substitute the calculated profit shares and given time periods:

₹3,725 : ₹2,980 = (C_M × 7 months) : (C_N × 6 months)

Simplify the profit ratio ₹3,725 : ₹2,980. Both numbers are divisible by 149:

3725 / 149 = 5

2980 / 149 = 20. Wait, let's recheck division. 3725 / 5 = 745. 2980 / 5 = 596. 745 = 5 * 149. 596 = 4 * 149. So the ratio is 5:4.

The simplified profit ratio is 5 : 4.

Now, we equate this to the capital and time ratio:

(C_M × 7) : (C_N × 6) = 5 : 4

This can be expressed as the equation:

\(\frac{7 C_M}{6 C_N} = \frac{5}{4}\)

Cross-multiply to solve for the relationship between C_M and C_N:

4 × (7 C_M) = 5 × (6 C_N)

28 C_M = 30 C_N

Simplify the equation by dividing both sides by 2:

14 C_M = 15 C_N

Finding M's Capital

We are given the relationship between M's and N's capital investments:

C_M = C_N + ₹79,000

Substitute this expression for C_M into the equation 14 C_M = 15 C_N:

14 (C_N + ₹79,000) = 15 C_N

Distribute the 14 on the left side:

14 C_N + (14 × ₹79,000) = 15 C_N

Calculate the product 14 × ₹79,000:

14 × 79,000 = 1,106,000

The equation becomes:

14 C_N + ₹1,106,000 = 15 C_N

Rearrange the terms to solve for C_N:

₹1,106,000 = 15 C_N - 14 C_N

C_N = ₹1,106,000

This result shows that N's capital contribution is ₹11,06,000.

Finally, calculate M's capital contribution using the given condition C_M = C_N + ₹79,000:

C_M = ₹1,106,000 + ₹79,000

C_M = ₹1,185,000

Thus, the capital contributed by M is ₹11,85,000.