Q.A invests $10000$ rupees for $6$ months and B invests $8000$ rupees for $9$ months. If the total profit is $8800$ rupees, find each partner's share.

Different times mean profit is shared in the ratio of capital x time.

One part is worth $8800/11 = 800$ rupees, and each partner takes their number of parts.

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

Step 1 — capital-months. $A: 10000 \times 6 = 60000, \qquad B: 8000 \times 9 = 72000.$

Step 2 — form and reduce the ratio. $60000 : 72000 = 5 : 6 \quad (\text{dividing both by } 12000).$

Step 3 — value of one part. Total parts $= 5 + 6 = 11$, so $\text{one part} = \frac{8800}{11} = 800 \text{ rupees}.$

Step 4 — each share. $A = 5 \times 800 = 4000, \qquad B = 6 \times 800 = 4800 \text{ rupees}.$

Verification. $4000 + 4800 = 8800$ rupees (the full profit), and $4000 : 4800 = 5 : 6$ matches the capital-time ratio — both confirm the answer.

As fractions: with capital-months summing to $132000$, $A = \frac{60000}{132000}\times 8800 = 4000$ and $B = \frac{72000}{132000}\times 8800 = 4800$ rupees.

Notice that although A invests the larger capital, B receives the larger share — because B's money was invested for longer. Ignoring the time and sharing by capital alone (which would favour A) is the classic error here.