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.
3 : 3 : 5
Let the initial capitals be 2k, 3k, 5k.
X's capital-months (unchanged for all 18 months):
\(X = 2k \times 18 = 36k\)
Y's capital-months: 3k for 6 months, then 1.5k for 12 months.
\(Y = 3k \times 6 + 1.5k \times 12 = 18k + 18k = 36k\)
Z's capital-months: 5k for 3 months, then 0.6 × 5k = 3k for 15 months.
\(Z = 5k \times 3 + 3k \times 15 = 15k + 45k = 60k\)
Ratio of profits:
\(X : Y : Z = 36k : 36k : 60k = 3 : 3 : 5\)
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