A and B enter into a partnership. A puts in $\text{ ₹ }$2000 but at the end of 3 months, withdraws, $\text{ ₹ }$500 and again at the end of 8 months withdraws $\text{ ₹ }$ 300. Out of a total profit of$\text{ ₹ }$ 900 at the end of the year, B's share was $\text{ ₹ }$400. Find B's capital.

Complete Solution:

A's investment changes during the year, so we calculate A's contribution in terms of "capital months" (investment × duration in months):

• First 3 months: A invested ₹2000.
  Capital months = 2000 × 3 = 6000.
• Next 5 months: After withdrawing ₹500, A invested ₹1500.
  Capital months = 1500 × 5 = 7500.
• Last 4 months: After withdrawing ₹300 more, A invested ₹1200.
  Capital months = 1200 × 4 = 4800.

Total capital months for A: 6000 + 7500 + 4800 = 18300.

Let B's capital be x. Since B's investment remains constant throughout the year:

B's capital months = 12x.

The profit is divided in the ratio of their effective capital contributions:

A's share : B's share = A's capital months : B's capital months.

Substituting the values:

18300 : 12x.

We know B's share of the profit is ₹400 out of ₹900. Thus, the profit-sharing ratio is:

(B's share / Total profit) = 4/9.

Set up the equation:

(12x) / (18300 + 12x) = 4 / 9.

Cross-multiply to simplify:

9 × 12x = 4 × (18300 + 12x)

108x = 73200 + 48x

108x - 48x = 73200

60x = 73200

x = 73200 / 60 = 1220.

Final Answer:

B's capital is ₹1220.