A and B together can complete a piece of work in 30 days, B and C together can complete it in 24 days, and C and A together can complete it in 20 days. All three work together for 10 days and then B and C leave. How many additional days will A alone take to finish the remaining work?

Introduction / Context:
This question is a classic example of using pairwise work rates to determine individual efficiencies. Three workers A, B and C have known times in pairs, and you must find how long A alone will need to complete the remaining work after a joint working period.

Given Data / Assumptions:

• A + B together finish the work in 30 days.
• B + C together finish in 24 days.
• C + A together finish in 20 days.
• All three work together for 10 days.
• After 10 days, only A continues working.
• Each worker's daily work rate is constant.

Concept / Approach:
Let a, b and c be daily work rates of A, B and C. Use the three given pairwise equations to solve for a, b and c. Then find the combined rate a + b + c, calculate how much work is finished in 10 days, and finally compute how long A alone, at rate a, takes to complete the remainder.

Step-by-Step Solution:
Let total work = 1 unit for convenience. Then a + b = 1 / 30. b + c = 1 / 24. c + a = 1 / 20. Add all three: 2(a + b + c) = 1/30 + 1/24 + 1/20. Compute the right side: LCM of 30, 24, 20 is 120. 1/30 = 4/120, 1/24 = 5/120, 1/20 = 6/120, sum = 15/120 = 1/8. So 2(a + b + c) = 1/8, hence a + b + c = 1/16. Now find A's rate: a = (a + b + c) - (b + c) = 1/16 - 1/24. 1/16 - 1/24 = (3 - 2) / 48 = 1 / 48. Thus A's rate = 1/48 of work per day. In 10 days together, work done = 10 * (1/16) = 10/16 = 5/8. Remaining work = 1 - 5/8 = 3/8. Time for A alone to finish remaining = (3/8) / (1/48) = 3/8 * 48 = 18 days.
Verification / Alternative check:
Check that individual rates are consistent: b = (a + b) - a = 1/30 - 1/48 and c = (b + c) - b; all remain positive. The remaining work fraction is less than half, so a time less than A's full 48 days is expected; 18 days is reasonable.
Why Other Options Are Wrong:
24, 30 or 36 days would imply A is much slower than computed and would contradict the earlier pairwise information. They also do not satisfy the proportion of remaining work 3/8 with rate 1/48.
Common Pitfalls:
Many aspirants forget to divide by 2 when adding the three pair equations. Some also treat total work as 100 or 120 without correctly normalizing the rates, which is fine only if done consistently. Always handle pair equations systematically and double check fraction arithmetic.
Final Answer:
A will take 18 days more to finish the work alone.