Two persons X and Y invested ₹ 15,000 and ₹ 20,000, respectively, in a…

Concept: In a partnership the profit is divided in the ratio of each partner’s capital multiplied by the time for which that capital stayed in the business; the capitals alone fix the ratio only in the special case where every partner’s money stays in for the same length of time. Writing that product as “capital-months”, the rule is Profit of first partner : Profit of second partner = (capital of first × time of first) : (capital of second × time of second).

Application: Apply that rule to the two investments in this question.

1. X keeps ₹15,000 in the business for the whole year, so X’s capital-months = 15,000 × 12 = 1,80,000.

2. Let Y’s money stay for t months, so Y’s capital-months = 20,000 × t.

3. The profit is shared 3 : 1 in the order X : Y, so 1,80,000 : 20,000t = 3 : 1.

4. Cross-multiplying gives 1,80,000 × 1 = 3 × 20,000t, that is 1,80,000 = 60,000t.

5. Dividing both sides by 60,000 gives t = 3.

Cross-check:

• Substituting back, X contributes 15,000 × 12 = 1,80,000 and Y contributes 20,000 × 3 = 60,000; the ratio 1,80,000 : 60,000 simplifies to 3 : 1, which is exactly the profit ratio given in the question.

• Sense check: Y put in the larger sum (₹20,000 against ₹15,000) yet takes the smaller share of the profit, so Y’s money must have stayed in the business for far less than a full year.

Hence Y invested his money for 3 months.