Q.A invests $12000$ rupees for the whole year. B joins the same business $3$ months later, investing $18000$ rupees for the remaining part of the year. If the profit for the year is $8500$ rupees, find each partner's share.

B's money is invested for only $9$ months, so the profit is shared in the ratio of each partner's capital x time.

B joins $3$ months after the start, so B's capital is invested for only $12 - 3 = 9$ months, while A's is invested for the full $12$ months. This is a compound partnership, and profit is shared in the ratio of capital × time.

Step 1 — capital-months. $A: 12000 \times 12 = 144000, \qquad B: 18000 \times 9 = 162000.$

Step 2 — form and reduce the ratio. $144000 : 162000 = 8 : 9 \quad (\text{dividing both by } 18000).$

Step 3 — value of one part. Total parts $= 8 + 9 = 17$, so $\text{one part} = \frac{8500}{17} = 500 \text{ rupees}.$ …

As fractions: with capital-months summing to $306000$, $A = \frac{144000}{306000}\times 8500 = 4000$ and $B = …

The most common mistake is using $12$ months for B instead of $9$ — when a partner joins late, their time runs only from the date they join to the end of the pe …