A, B and C can complete a piece of work in 24 days, 30 days and 40 days respectively when working alone. They begin the work together, but C leaves 4 days before the completion of the work. In how many days was the entire work completed?

Introduction / Context:
This time and work problem involves three workers with different efficiencies. All start together, but one worker (C) leaves before the work is over. The question checks if you can translate this situation into a simple equation using daily work rates and total work units.

Given Data / Assumptions:

• A alone finishes in 24 days.
• B alone finishes in 30 days.
• C alone finishes in 40 days.
• All three start together.
• C leaves 4 days before completion.
• We assume constant daily work rates.

Concept / Approach:
Use the idea of work as units. Choose a convenient total work value as the least common multiple of 24, 30 and 40 so that each person's daily work rate is an integer. Compute total work contributed by each person up to completion, using the fact that C works D - 4 days if D is total days taken.

Step-by-Step Solution:
Let total work = LCM of 24, 30 and 40 = 120 units. A's rate = 120 / 24 = 5 units per day. B's rate = 120 / 30 = 4 units per day. C's rate = 120 / 40 = 3 units per day. Let the total time taken be D days. A works D days and does 5D units. B works D days and does 4D units. C leaves 4 days early, so works D - 4 days and does 3(D - 4) units. Total work equation: 5D + 4D + 3(D - 4) = 120. Simplify: 9D + 3D - 12 = 120, so 12D - 12 = 120. Therefore 12D = 132 and D = 132 / 12 = 11 days.
Verification / Alternative check:
In 11 days A does 5 * 11 = 55 units. B does 4 * 11 = 44 units. C works 7 days and does 3 * 7 = 21 units. Total = 55 + 44 + 21 = 120 units, which matches the total work.
Why Other Options Are Wrong:
12 days would yield larger total work than 120 units using the same rates. 13 or 14 days would increase work even more, so the workers would overshoot the required job by a large margin. Thus only 11 days satisfies the exact work requirement.
Common Pitfalls:
Students sometimes forget that C works fewer days, mistakenly using D instead of D - 4 for C. Another error is computing LCM or daily work rates incorrectly, which carries as a mistake through the equation. Always write a clear expression for total work and check the arithmetic carefully.
Final Answer:
The work is completed in 11 days.