A and B start a business together. The amount invested by A is three times the amount invested by B, and A keeps his investment in the business for twice as long as B. If B's share of the profit is Rs. 4,000, what is their total profit?

Introduction / Context:
This partnership question involves both differing investment amounts and differing periods of investment. The combined effect of capital and time determines each partner's share of profit. We are given B's profit share and need to find the total profit earned by the business, taking the effective investments into account.

Given Data / Assumptions:

• A and B are partners in a business.
• Investment of A is three times the investment of B.
• Time period of A's investment is twice the time period of B's investment.
• B's share of the profit is Rs. 4,000.
• Profit is shared in proportion to capital multiplied by time.
• We must determine the total profit of the business.

Concept / Approach:
In partnership problems, the effective investment is capital * time. We express A's and B's effective investments as multiples of a common base. This directly gives us the ratio of their profit shares. Knowing B's share and the ratio, we can find the total profit by scaling up the ratio parts to the actual rupee amounts.

Step-by-Step Solution:
Step 1: Let B's capital be x rupees and his investment period be t months. Step 2: Then A's capital is 3x and A's time period is 2t months. Step 3: Effective investment of A = 3x * 2t = 6xt. Step 4: Effective investment of B = x * t = xt. Step 5: Therefore the ratio of profit shares of A and B = 6xt : xt = 6 : 1. Step 6: Let the total profit be P. Then B's share corresponds to 1 part out of 7 parts (6 for A and 1 for B). Step 7: Since B's share (1 part) is Rs. 4,000, total profit P = 7 * 4,000 = Rs. 28,000. Step 8: A's share would then be 6 * 4,000 = Rs. 24,000, which is consistent with the ratio.

Verification / Alternative check:
We can verify quickly: If the profit ratio is 6 : 1 and total profit is 28,000, then B gets 1/7 of 28,000 which is 4,000, matching the given data. A gets the remaining 24,000. This makes the total profit 24,000 + 4,000 = 28,000, confirming that the calculation is correct.

Why Other Options Are Wrong:
If the total profit were Rs. 22,000, Rs. 32,000 or Rs. 36,000, then B's share computed as one seventh of the total would be approximately 3,143, 4,571 or 5,143 respectively. None of these values equal the given B's share of Rs. 4,000, so those options do not satisfy the condition of the question.

Common Pitfalls:
Learners sometimes forget to multiply capital by time and instead compare only the capitals or only the time periods. Another frequent error is to assume that the profit is divided in a simple ratio like 3 : 1 without considering the effect of A investing for twice the time. Always treat partnership profit shares as proportional to capital * time, especially when the time periods are different.

Final Answer:
The total profit of the business is Rs. 28,000.