Q.A, B and C invest $4000$ rupees, $5000$ rupees and $6000$ rupees for $6$ months, $8$ months and $4$ months respectively. If the total profit is $5500$ rupees, find each partner's share.

With three different capitals and three different times, profit is shared in the ratio of the three capital x time products.

The times differ, so this is a compound partnership; profit is shared in the ratio of capital × time.

Step 1 — capital-months. $A: 4000 \times 6 = 24000, \quad B: 5000 \times 8 = 40000, \quad C: 6000 \times 4 = 24000.$

Step 2 — form and reduce the ratio. $24000 : 40000 : 24000 = 3 : 5 : 3 \quad (\text{dividing each by } 8000).$

Step 3 — value of one part. Total parts $= 3 + 5 + 3 = 11$, so $\text{one part} = \frac{5500}{11} = 500 \text{ rupees}.$

Step 4 — each share. $A = 3 \times 500 = 1500, \quad B = 5 \times 500 = 2500, \quad C = 3 \times 500 = 1500 \text{ rupees}.$ …

As fractions of the total: with the capital-months summing to $88000$, $A = \frac{24000}{88000}\times 5500 = 1500$, $B = \frac{40000}{88000}\times 5 …

A common error is to assume equal capital-months means equal capital or equal time — here A and C have neither, yet share equally, because it …