Solving Business Partnership Profit Sharing Problems

This problem involves calculating the capital invested by partners in a business, based on their investment duration and how the total profit is shared. In a partnership, profits are generally distributed among partners in the ratio of the product of their respective capital investments and the time period for which the capital was invested.

Understanding the Partnership Details

We are given the following information about partners P and Q:

• P invests ₹14,000 more than Q.
• P invests for 8 months.
• Q invests for 10 months.
• Total profit is ₹2,000.
• P's share of the profit is ₹400 more than Q's share.

We need to find the capital contributed by P.

Calculating Individual Profit Shares

Let \(S_P\) be P's share of the profit and \(S_Q\) be Q's share of the profit.

According to the problem:

• The total profit is ₹2,000, so \(S_P + S_Q = 2000\).
• P's share is ₹400 more than Q's share, so \(S_P = S_Q + 400\).

Now we can substitute the second equation into the first one:

\((S_Q + 400) + S_Q = 2000\)

\(2S_Q + 400 = 2000\)

\(2S_Q = 2000 - 400\)

\(2S_Q = 1600\)

\(S_Q = \frac{1600}{2}\)

\(S_Q = 800\)

Now we find \(S_P\):

\(S_P = S_Q + 400\)

\(S_P = 800 + 400\)

\(S_P = 1200\)

So, P's profit share is ₹1,200 and Q's profit share is ₹800. The ratio of their profit shares is \(S_P : S_Q = 1200 : 800\), which simplifies to \(12 : 8\), and further simplifies to \(3 : 2\).

Relating Profit Shares to Capital and Time

The ratio of profit shares is equal to the ratio of (Capital \(\times\) Time) for each partner. Let \(C_P\) be P's capital and \(C_Q\) be Q's capital.

We know that P puts ₹14,000 more than Q, so \(C_P = C_Q + 14000\).

The time periods are \(T_P = 8\) months and \(T_Q = 10\) months.

The ratio of (Capital \(\times\) Time) is \((C_P \times T_P) : (C_Q \times T_Q)\). This must be equal to the profit share ratio \(S_P : S_Q\).

\(\frac{C_P \times T_P}{C_Q \times T_Q} = \frac{S_P}{S_Q}\)

Substitute the values we know:

\(\frac{(C_Q + 14000) \times 8}{C_Q \times 10} = \frac{1200}{800}\)

Simplify the ratio on the right side:

\(\frac{8(C_Q + 14000)}{10C_Q} = \frac{3}{2}\)

Further simplify the left side by dividing 8 and 10 by 2:

\(\frac{4(C_Q + 14000)}{5C_Q} = \frac{3}{2}\)

Solving for Capital Investment

Now, we solve the equation for \(C_Q\) using cross-multiplication:

\(2 \times 4(C_Q + 14000) = 3 \times 5C_Q\)

\(8(C_Q + 14000) = 15C_Q\)

Distribute the 8 on the left side:

\(8C_Q + 8 \times 14000 = 15C_Q\)

\(8C_Q + 112000 = 15C_Q\)

Subtract \(8C_Q\) from both sides to isolate \(C_Q\):

\(112000 = 15C_Q - 8C_Q\)

\(112000 = 7C_Q\)

Divide by 7 to find \(C_Q\):

\(C_Q = \frac{112000}{7}\)

\(C_Q = 16000\)

So, Q's capital contribution is ₹16,000.

The question asks for the capital contributed by P. We know that \(C_P = C_Q + 14000\).

\(C_P = 16000 + 14000\)

\(C_P = 30000\)

Therefore, the capital contributed by P is ₹30,000.

Checking the Answer

Let's quickly verify if this works out:

• P's capital = ₹30,000, Time = 8 months. Product = \(30000 \times 8 = 240000\).
• Q's capital = ₹16,000, Time = 10 months. Product = \(16000 \times 10 = 160000\).
• Ratio of products = \(240000 : 160000 = 24 : 16 = 3 : 2\).

The total profit is ₹2,000. If this is shared in the ratio 3:2, the total ratio parts are \(3 + 2 = 5\).

• Value per ratio part = \(\frac{2000}{5} = 400\).
• P's share = \(3 \times 400 = 1200\).
• Q's share = \(2 \times 400 = 800\).

Is P's share ₹400 more than Q's share? \(1200 - 800 = 400\). Yes, it is.

The calculated capital amounts correctly satisfy all conditions in the problem.

Revision Table: Business Partnership Basics

Additional Information on Partnership Problems

Partnership problems are common in quantitative aptitude sections of various exams. They typically involve calculating capitals, time periods, or profit shares based on given ratios and total amounts. The core principle is that the profit earned by a partner is proportional to their effective investment, which is the product of the amount invested and the time it was invested for.

Key points to remember:

• Always ensure the units for time (months, years, etc.) are consistent for all partners before calculating the (Capital \(\times\) Time) product.
• If a partner receives a salary or commission from the profit before distribution, this is usually mentioned explicitly and handled separately. The remaining profit is then distributed based on the capital-time ratio.
• Sometimes, a working partner might receive a higher share of profit or a fixed amount in addition to their share based on investment.

Practicing various types of partnership problems helps in understanding how different conditions (like additional investments, withdrawals, or varying time periods) affect the profit distribution.