A, B and C enter into a partnership by investing Rs. 40,000, Rs. 80,000 and Rs. 1,20,000 respectively. At the end of the first year, B withdraws Rs. 40,000, and at the end of the second year, C withdraws Rs. 80,000. If the profit is calculated at the end of 3 years, in what ratio will it be shared among A, B and C?

Introduction / Context:
This problem involves a three year partnership in which two partners partially withdraw capital at different stages. Because capital amounts change from year to year, we must compute the effective investment of each partner in terms of capital * time before determining the final profit sharing ratio between A, B and C.

Given Data / Assumptions:

• Initial investments: A = Rs. 40,000, B = Rs. 80,000, C = Rs. 1,20,000.
• The business runs for a total of 3 years.
• At the end of year 1, B withdraws Rs. 40,000, so in years 2 and 3 B invests Rs. 40,000.
• At the end of year 2, C withdraws Rs. 80,000, so in year 3 C invests Rs. 40,000.
• A keeps his full investment unchanged for the entire 3 years.
• Profit is shared in proportion to capital * time.

Concept / Approach:
We split the 3 year period into separate one year blocks. For each year we list the capitals of A, B and C and compute their effective investments by multiplying each capital by the time span of that block. Summing the effective investments over all three years gives us total effective investments, and their ratio gives the profit sharing ratio at the end of 3 years.

Step-by-Step Solution:
Step 1: Year 1: A invests 40,000, B invests 80,000, C invests 1,20,000 for 1 year each. Effective investments: A = 40,000 * 1 = 40,000, B = 80,000 * 1 = 80,000, C = 1,20,000 * 1 = 1,20,000. Step 2: Year 2: B withdraws 40,000, so: A = 40,000, B = 40,000, C = 1,20,000 for 1 year. Effective investments: A = 40,000, B = 40,000, C = 1,20,000. Step 3: Year 3: C withdraws 80,000, so: A = 40,000, B = 40,000, C = 40,000 for 1 year. Effective investments: A = 40,000, B = 40,000, C = 40,000. Step 4: Total effective investments across 3 years: A = 40,000 + 40,000 + 40,000 = 1,20,000, B = 80,000 + 40,000 + 40,000 = 1,60,000, C = 1,20,000 + 1,20,000 + 40,000 = 2,80,000. Step 5: To find the ratio, divide all by 40,000: A : B : C = 3 : 4 : 7. Step 6: Therefore, profit will be shared in the ratio 3 : 4 : 7 among A, B and C.

Verification / Alternative check:
We can verify consistency by imagining a sample profit amount, for example Rs. 1,40,000, and dividing it in the ratio 3 : 4 : 7. The shares would be 30,000, 40,000 and 70,000 respectively, and the relative magnitudes match the overall pattern of investments, where C had the largest effective investment, B was in the middle and A had the smallest. This confirms that the ratio 3 : 4 : 7 is sensible and consistent with the capital history.

Why Other Options Are Wrong:
Options 2 : 3 : 5, 5 : 6 : 4 and 1 : 3 : 5 do not match the effective investment totals. For example, A and B have equal effective investments in the last year but B invested more in the first year, so any correct ratio must have B's final share greater than A's. Ratios that do not reflect this ordering or do not correspond to 1,20,000 : 1,60,000 : 2,80,000 cannot be correct.

Common Pitfalls:
Students may incorrectly use only initial capitals or only final capitals and ignore the timing of withdrawals. Another error is to simply average capital amounts instead of multiplying by time. Always track each capital change with its duration and sum the capital * time values to get accurate ratios in multi year partnership questions.

Final Answer:
The profit will be shared in the ratio 3 : 4 : 7 among A, B and C.