X and Y invest in a business and earn a profit which they divide in the ratio 2 : 3. If X invests Rs. 40,000, what amount of capital must have been invested by Y?

Introduction / Context:
This is a straightforward partnership question. Two partners X and Y share profit in a known ratio. We are given the capital invested by X and must find the capital invested by Y, assuming that both partners kept their money in the business for the same length of time.

Given Data / Assumptions:

• Profit sharing ratio of X : Y is 2 : 3.
• X invested Rs. 40,000.
• The duration of investment for both X and Y is the same.
• Profit is shared in proportion to capital, since time is equal.
• We need to find Y's capital investment.

Concept / Approach:
If the time for which the money is invested is the same for both partners, then the profit share ratio is simply the ratio of their capitals. Therefore, capital of X : capital of Y = 2 : 3. Knowing X's capital and this ratio, we can find Y's capital by a simple proportion calculation.

Step-by-Step Solution:
Step 1: Write the profit sharing ratio as capital ratio: capital of X : capital of Y = 2 : 3. Step 2: X's capital corresponds to the first part in the ratio. So 2 parts represent Rs. 40,000. Step 3: Value of 1 part = 40,000 / 2 = Rs. 20,000. Step 4: Y's capital corresponds to 3 parts. So Y's capital = 3 * 20,000 = Rs. 60,000. Step 5: Therefore, Y must have invested Rs. 60,000 to justify the 2 : 3 profit sharing ratio.

Verification / Alternative check:
If X invests 40,000 and Y invests 60,000, then the capital ratio is 40,000 : 60,000. Dividing each by 20,000, we get 2 : 3, which matches the given profit ratio. This confirms that our proportion method is correct and that Y's capital has been correctly computed.

Why Other Options Are Wrong:
If Y had invested Rs. 20,000, Rs. 40,000 or Rs. 80,000, the resulting capital ratios with X's 40,000 would be 2 : 1, 1 : 1 or 1 : 2 respectively. None of these ratios equal the required profit sharing ratio of 2 : 3. Hence those options cannot give the correct relationship between the partners' profit shares.

Common Pitfalls:
A common mistake is to divide the capital in the inverse ratio or to assume that Y's investment must be less than X's. Another error is to forget that when time is the same, the profit ratio directly equals the capital ratio, with no need to consider capital * time. Always apply the given ratio directly to the capitals when time does not differ.

Final Answer:
Y invested a capital amount of Rs. 60,000.