M and N start a business with Rs. 2,00,000 and Rs. 3,00,000 respectively. After 4 months, M adds Rs. 1,00,000 more, and N withdraws Rs. 50,000. At the end of 1 year, they make a profit of Rs. 1,80,000. What is N’s share of profit?

Correct — B. Profit is shared in the ratio of capital × months.

M: ₹2,00,000 for 4 months, then ₹3,00,000 for 8 months
= 8,00,000 + 24,00,000 = 32,00,000
N: ₹3,00,000 for 4 months, then ₹2,50,000 for 8 months
= 12,00,000 + 20,00,000 = 32,00,000

Ratio M : N = 1 : 1
N's share = ₹1,80,000 ÷ 2 = ₹90,000 → option (b)

Why the others are wrong

• (a)
  Rs. 96,000 — Rs. 96,000 leaves M ₹84,000, a split of 8 : 7 in N's favour. The capital-months are equal, so the split must be even.
• (c)
  Rs. 84,000 — Rs. 84,000 gives N the smaller part, 7 : 8. N's capital-months equal M's at 32,00,000, so neither partner gets less.
• (d)
  Rs. 88,000 — Rs. 88,000 splits the profit 22 : 23 against N. Equal capital-months of 32,00,000 each mean ₹90,000 apiece.

When partners stay for different lengths of time, or change their capital midway, compare capital × time, not capital alone.

Split the year at each change. Here the change comes after 4 months, so each partner has a 4-month stretch and an 8-month stretch.

The two leads cancel. N is ₹1,00,000 ahead for the first 4 months (4,00,000) and M is ₹50,000 ahead for the last 8 (also 4,00,000), so the ratio is 1 : 1.

Key facts

• Profit ratio = ratio of each partner's capital × months.
• A mid-year change splits a partner's investment into separate periods, each multiplied by its own months.
• Equal capital-months mean equal shares, whatever the starting amounts.

Common traps

• Using only the starting capitals, 2 : 3: that gives N ₹1,08,000, which is not an option.
• Using the later capitals for the whole year, 3,00,000 : 2,50,000 = 6 : 5, which gives N about ₹81,818.

No directly related past PYQ was found.