A and B invest Rs. 1,16,000 and Rs. 1,44,000 respectively in a business. B keeps his capital invested for 6 months, while A invests for the full year. At the end of the year, B's share of profit is Rs. 9000. What is the total profit earned by the business?

Introduction / Context:
This partnership problem involves different capitals and different time periods of investment. The question gives the profit share of one partner and asks for the total profit. To solve this, we use the idea that profit shares are proportional to the products of capital and time for each partner.

Given Data / Assumptions:

• A invests Rs. 1,16,000 for 12 months.
• B invests Rs. 1,44,000 for 6 months.
• B's share of profit at the end of the year is Rs. 9000.
• We need to find the total profit earned in the business.

Concept / Approach:
In partnership questions, each partner's share is proportional to capital * time. That means we first calculate the effective capital time products and then express them as a ratio. Knowing one partner's actual rupee share along with the ratio enables us to scale up and find the total profit. This avoids any need to know the exact total capital.

Step-by-Step Solution:
Step 1: Calculate A's capital time product: 1,16,000 * 12 = 13,92,000. Step 2: Calculate B's capital time product: 1,44,000 * 6 = 8,64,000. Step 3: The ratio of A's share to B's share is 13,92,000 : 8,64,000. Step 4: Simplify this ratio by dividing both terms by 48,000, giving 29 : 18. Step 5: So the profit ratio A : B is 29 : 18. Step 6: Total number of parts in the ratio is 29 + 18 = 47. Step 7: B gets 18 parts out of 47, and we know these 18 parts are worth Rs. 9000. Step 8: Value of one part = 9000 / 18 = 500. Step 9: Total profit = 47 * 500 = Rs. 23,500.

Verification / Alternative check:
If the total profit is Rs. 23,500 and one part is Rs. 500, then A's share is 29 parts = 29 * 500 = Rs. 14,500. B's share is 18 parts = 18 * 500 = Rs. 9000, which matches the given value. Adding the two shares, 14,500 + 9000 = 23,500, confirms that the total profit is consistent with the derived ratio and the information provided in the question.

Why Other Options Are Wrong:
Rs. 19,780, Rs. 21,700 and Rs. 20,050 do not yield a per part value that gives B's share exactly equal to Rs. 9000 while maintaining the ratio 29 : 18. When tested, they either give non integer values per part or give a B share different from Rs. 9000, so they must be rejected.

Common Pitfalls:
A usual mistake is to ignore the time periods and compare only capitals, which gives a wrong ratio. Another error is to miscalculate the simplification of the ratio or to forget to add the ratio parts before scaling to the total profit. Always compute capital time products first, then simplify the ratio carefully and finally convert ratio parts into money values.

Final Answer:
The total profit earned by the business is Rs. 23,500.