Sohan started a business with ₹90,000. After 5 months, Rohan joined with ₹1,50,000. After another 3 months, Sohan withdrew ₹30,000. What is the ratio of their profits at the end of the year?

Correct — A. Profit splits in the ratio of capital × months.

Rohan joins at the end of month 5
Sohan withdraws 3 months later, at the end of month 8

Sohan, months 1–8: 90,000 × 8 = 7,20,000
Sohan, months 9–12: (90,000 − 30,000) × 4 = 2,40,000
Add: Sohan = 9,60,000

Rohan, months 6–12: 1,50,000 × 7 = 10,50,000

Ratio: 9,60,000 : 10,50,000 = 96 : 105
Divide by 3: 32 : 35 → option (a)

Why the others are wrong

• (b)
  15:16 — 15 : 16 is not 96 : 105 in any form. 96 and 105 share only the factor 3, so the ratio reduces to 32 : 35 and no further.
• (c)
  12:15 — 12 : 15 reduces to 4 : 5 = 0.8, well below the true 9,60,000 ⁄ 10,50,000 ≈ 0.914.
• (d)
  11:12 — 11 : 12 ≈ 0.917 is close but not equal. Cross-multiplying, 32 × 12 = 384 while 35 × 11 = 385, so 11 : 12 is not 32 : 35.

In a partnership with no salary or interest, profit is shared in the ratio of capital × time. When a partner's capital changes, split that partner's year into periods and add capital × months for each.

Mark every event on a 12-month line first. 'After another 3 months' counts from the previous event, so Sohan's cut comes at month 8, not month 3.

Key facts

• With no salary or interest, profits divide in the ratio of capital × months.
• A partner who joins after 5 months of a 12-month year is in for 7 months.
• 9,60,000 : 10,50,000 = 96 : 105 = 32 : 35.

Common traps

• Timing the withdrawal from the start: 90,000 × 3 + 60,000 × 9 = 8,10,000 against 10,50,000 gives 27 : 35.
• Ignoring the withdrawal: 90,000 × 12 = 10,80,000 against 10,50,000 gives 36 : 35.

No directly related past PYQ was found.