5.2 Compound Partnership (different time periods)

When partners invest for different durations, profit is divided in the ratio of (Investment × Time) for each partner — this product is called “capital-months” (or capital-days).

Formula: Profit share ratio = (C₁ × T₁) : (C₂ × T₂) : …

Example 32: A invests ₹4000 for 6 months and B invests ₹6000 for 4 months. Divide a profit of ₹5000 between them. A’s capital-months = 4000 × 6 = 24000 B’s capital-months = 6000 × 4 = 24000 Ratio = 24000 : 24000 = 1 : 1 → Each gets ₹2500.

Example 33: A starts a business with ₹8000. After 4 months, B joins with ₹12000. At the end of the year, the profit is ₹7500. Find each partner’s share. A invests for the full 12 months: 8000 × 12 = 96000 B invests for the remaining 8 months (12 − 4): 12000 × 8 = 96000 Ratio = 96000 : 96000 = 1 : 1 → A = B = ₹3750 each.

Example 34 (with mid-year change — a favourite CGL-level twist): A and B start a business investing ₹35,000 and ₹45,000 respectively. After 4 months, A withdraws ₹5000 and B invests an additional ₹5000. Find the profit-sharing ratio at the end of the year. A’s capital-months = (35000 × 4) + (30000 × 8) = 140000 + 240000 = 380000 B’s capital-months = (45000 × 4) + (50000 × 8) = 180000 + 400000 = 580000 Ratio = 380000 : 580000 = 38 : 58 = 19 : 29 (dividing by 2).