A started a business with a capital of ₹8,42,000. After 7 months, B joined him…

Concept

In a partnership where partners invest different capitals for different durations, profit is shared not in the ratio of the capitals alone but in the ratio of each partner's capital multiplied by the number of months it remained invested — that is, in the ratio of their investment-months (capital × time).

Application

1. A's ₹8,42,000 stayed invested for the whole year, i.e., for 12 months, so A's investment-months = 8,42,000 × 12 = 1,01,04,000.

2. B joined 7 months after the business started, so B's capital (let it be ₹x) remained invested for only the remaining 12 − 7 = 5 months, giving B's investment-months = 5x.

3. The profit was divided in the ratio 6 : 2, which simplifies to 3 : 1, so A's and B's investment-months must also be in the ratio 3 : 1: 1,01,04,000 : 5x = 3 : 1.

4. Cross-multiplying, 1 × 1,01,04,000 = 3 × 5x, so 15x = 1,01,04,000, giving x = 1,01,04,000 ÷ 15 = ₹6,73,600.

Cross-check

With x = ₹6,73,600, B's investment-months = 5 × 6,73,600 = 33,68,000, and A's investment-months = 1,01,04,000. The ratio 1,01,04,000 : 33,68,000 reduces to 3 : 1, i.e., 6 : 2 — matching the profit split given in the question, so B's investment of ₹6,73,600 checks out.