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?

The correct option is ₹ 30,000.

Let us break down the solution step-by-step to understand how we arrive at this answer.

Step 1: Understand the division of profit
The total profit from the business is ₹ 2,000.
Let the profit share of Q be ₹ x.
According to the question, P's profit share is ₹ 400 more than Q's share. Therefore, P's profit share is ₹ (x + 400).

Since the total profit is the sum of the profit shares of P and Q, we can write:
$P\text{'}s \text{share}+Q\text{'}s \text{share}=\text{Total Profit}$

$(x+400)+x=2000$

$2x+400=2000$

$2x=1600$

$x=800$

So, Q's share of the profit is ₹ 800, and P's share of the profit is:
$800+400=1200$

Therefore, the ratio of the profits of P and Q is:
$\frac{\text{Profit of }P}{\text{Profit of }Q}=\frac{1200}{800}=\frac{3}{2}$

Step 2: Relate profit ratio to investment and time
In a partnership, the ratio of profit distribution is directly proportional to the product of the capital invested and the time period of investment.
$\frac{\text{Profit of }P}{\text{Profit of }Q}=\frac{\text{Capital of }P\times \text{Time of }P}{\text{Capital of }Q\times \text{Time of }Q}$

Let the capital contributed by Q be ₹ C_Q.
Since P contributes ₹ 14,000 more than Q, the capital contributed by P is ₹ (C_Q + 14,000).
The investment duration for P is 8 months, and for Q is 10 months.

Substituting these values into the ratio formula:
$\frac{3}{2}=\frac{(C_Q+14000)\times 8}{C_Q\times 10}$

Step 3: Solve for Q's capital (C_Q)
First, simplify the fraction on the right-hand side:
$\frac{3}{2}=\frac{4\times (C_Q+14000)}{5\times C_Q}$

Cross-multiplying to solve for C_Q:
$3\times 5C_Q=2\times 4\times (C_Q+14000)$

$15C_Q=8C_Q+112000$

Subtracting 8C_Q from both sides:
$7C_Q=112000$

$C_Q=\frac{112000}{7}$

$C_Q=16000$

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

Step 4: Find the capital contributed by P
P's capital is ₹ 14,000 more than Q's capital:
$\text{Capital of }P=C_Q+14000$

$\text{Capital of }P=16000+14000=30000$

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