Umar and Avinash start a partnership with investments of Rs. 10,000 and Rs. 15,000 respectively. Due to a financial emergency, Avinash withdraws his entire investment after 8 months, while Umar keeps his capital invested for the full 12 months. In what ratio should the profit of the first year be shared between Umar and Avinash?

Introduction / Context:
This is a partnership problem where partners invest different amounts for different durations. We must calculate the effective capital time for each partner and then determine the ratio in which the profit should be shared. Such questions are very common in aptitude tests and banking exams.

Given Data / Assumptions:

• Umar invests Rs. 10,000 at the beginning and keeps it for the full 12 months.
• Avinash invests Rs. 15,000 at the beginning but withdraws his amount after 8 months.
• Profit is shared in proportion to capital multiplied by time.
• The period under consideration is 12 months (one year).

Concept / Approach:
When partners change their investments during the course of a business, we do not simply compare their capital amounts. Instead, we compare the product of capital and time, often referred to as money months. Once we calculate the capital time contributions, the ratio of these products gives the ratio of their profit shares.

Step-by-Step Solution:
Step 1: Umar's investment is Rs. 10,000 for 12 months.Step 2: Umar's capital time = 10,000 * 12 = 1,20,000 money months.Step 3: Avinash's investment is Rs. 15,000 for 8 months.Step 4: Avinash's capital time = 15,000 * 8 = 1,20,000 money months.Step 5: The ratio of Umar : Avinash in terms of capital time is 1,20,000 : 1,20,000.Step 6: Simplify this ratio by dividing both sides by 1,20,000 to get 1 : 1.Step 7: Therefore, the profit of the first year should be shared equally between the two partners.

Verification / Alternative check:
We can interpret in another way: Avinash invests 1.5 times the capital of Umar (15,000 versus 10,000) but only for two thirds of the time (8 months versus 12 months). Multiplying these factors, 1.5 * (8/12) = 1.5 * 2/3 = 1. This shows that his overall contribution is exactly equal to that of Umar, confirming the 1 : 1 profit sharing ratio.

Why Other Options Are Wrong:
Ratios 3 : 2, 2 : 3 and 1 : 2 suggest unequal contributions that would arise only if one partner had a higher weighted capital time. However, both partners contribute equally when capital and time are combined. 'None of these' is not correct because one of the offered ratios, 1 : 1, matches the computed profit ratio exactly.

Common Pitfalls:
Students often look only at capital amounts or only at time and assume that the partner who invested more money receives more profit. In this problem, ignoring time or capital would lead to incorrect conclusions. Always use capital multiplied by time to compare profit shares in such partnership questions.

Final Answer:
The profit for the first year should be shared between Umar and Avinash in the ratio 1 : 1.