Two persons P and Q enter into a business. P puts ₹14,000 more than Q, but P has invested for 8 months and Q has invested for 10 months. If P’s share is ₹400 more than Q’s share out of the total profit of ₹2,000, what is the capital contributed by P?

Correct Option

Let P's capital be P and Q's capital be Q.

Based on the problem statement:

• P's capital is ₹14,000 more than Q's: P = Q + 14000, or Q = P – 14000.
• P invested for 8 months.
• Q invested for 10 months.
• Total profit = ₹2,000.
• P's share of profit (S_P) is ₹400 more than Q's share (S_Q).

First, determine the individual profit shares:

• The sum of profit shares is the total profit: S_P + S_Q = 2000.
• Given S_P = S_Q + 400.
• Substituting S_P into the sum equation: (S_Q + 400) + S_Q = 2000.
• This simplifies to 2S_Q = 1600, so S_Q = ₹800.
• Consequently, S_P = 800 + 400 = ₹1200.

The profit sharing ratio is directly proportional to the product of capital and time invested:

$\frac{\text{P’s Capital}\times \text{P’s Time}}{\text{Q’s Capital}\times \text{Q’s Time}}=\frac{\text{P’s Profit Share}}{\text{Q’s Profit Share}}$

Substitute the known values:

$\frac{P\times 8}{(P-14000)\times 10}=\frac{1200}{800}$

Simplify the profit ratio $\frac{1200}{800}=\frac{3}{2}$:

$\frac{8P}{10(P-14000)}=\frac{3}{2}$

Cross-multiply to solve for P:

$\Rightarrow$ $2\times 8P=3\times 10(P-14000)$

$\Rightarrow$ $16P=30(P-14000)$

$\Rightarrow$ $16P=30P-420000$

$\Rightarrow$ $420000=30P-16P$

$\Rightarrow$ $420000=14P$

$\Rightarrow$ $P=\frac{420000}{14}$

$\Rightarrow$ P = 30000

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

Incorrect Options

Options 2 (₹26,000), 3 (₹24,000), and 4 (₹20,000) are incorrect. These values for P's capital do not satisfy the conditions derived from the investment terms and the specified profit distribution between P and Q.