A started a business with Rs. 2,70,000 and was joined by B three months…

In partnership problems where partners invest different capital amounts for different durations, the profit is shared in proportion to each partner's capital multiplied by the time (in months) that the capital remained invested — not by capital alone. When the profit split is given as a fraction of the total, converting it into a ratio between the partners lets you equate it to the capital × time ratio and solve for an unknown investment.

Applying this to the given problem:

1. A invests Rs. 2,70,000 for the full year (12 months). B joins 3 months later, so B’s capital stays invested for 12 − 3 = 9 months. Let B’s investment be Rs. N.

2. The capital × time ratio for A and B is (2,70,000 × 12) : (N × 9).

3. A’s profit share is three-fifths of the total profit, so B’s share is the remaining two-fifths — the profit ratio A : B is therefore 3 : 2.

4. Equating the two ratios: (2,70,000 × 12) : (N × 9) = 3 : 2.

5. Cross-multiplying: 2,70,000 × 12 × 2 = 3 × N × 9, which gives 64,80,000 = 27N.

6. Solving for N: N = 64,80,000 ÷ 27 = 2,40,000.

Cross-check: with N = 2,40,000, A’s capital-time product is 2,70,000 × 12 = 32,40,000 and B’s is 2,40,000 × 9 = 21,60,000. The ratio 32,40,000 : 21,60,000 simplifies to 3 : 2, matching the given condition.

Hence, B invested Rs. 2,40,000.