X, Y, and Z started a business with capitals in the ratio 2:3:5. After 3 months, Z withdrew 40% of his capital. After another 3 months, Y withdrew half of his capital. The business continued for a total of 18 months. If total profit is ₹3,24,000, find the ratio of profit shares of X : Y : Z.

Correct — A. Take the capitals as 2, 3 and 5 units over 18 months.

Z withdraws 40% after month 3, keeping 5 × 0.6 = 3
Y withdraws half after month 6, keeping 3 ÷ 2 = 1.5

X: 2 × 18 = 36
Y: 3 × 6 + 1.5 × 12 = 18 + 18 = 36
Z: 5 × 3 + 3 × 15 = 15 + 45 = 60

Divide 36 : 36 : 60 by 12: 3 : 3 : 5 → option (a)

Why the others are wrong

• (b)
  3:3:4 — Z's capital-months are 60, not 48. 3 : 3 : 4 would need Z = 48, but 5 × 3 + 3 × 15 = 60, which is 5 parts of 12.
• (c)
  1:2:2 — 1 : 2 : 2 gives Y twice X's share. Halving Y's capital after month 6 leaves Y at 3 × 6 + 1.5 × 12 = 36, exactly X's 36.
• (d)
  1:3:1 — 1 : 3 : 1 makes Z equal to X, yet Z holds at least 3 units for all 18 months against X's 2. Z's share must be larger than X's.

Profit shares follow capital × time, added over each period in which a partner's capital stays fixed. Capitals given only as a ratio can be taken as that many units, since the answer is itself a ratio.

A withdrawal of p% leaves (100 − p)% for the remaining months. 'After another 3 months' is measured from the previous event, so Y's cut comes at month 6.

Key facts

• Partners' profit ratio = ratio of capital × months, summed over each period.
• Withdrawing 40% of a capital leaves 60% of it.
• 36 : 36 : 60 = 3 : 3 : 5, dividing by 12.

Common traps

• Keeping 40% after Z's withdrawal instead of 60%: 'withdrew 40%' leaves 5 × 0.6 = 3 units.
• Timing Y's withdrawal from the start: it comes 3 months after Z's, at month 6.
• Working the ₹3,24,000 into the ratio: the total only matters for a rupee share.

No directly related past PYQ was found.