X, Y and Z can complete a piece of work in 30, 40 and 50 days respectively when working alone. All three begin the work together, but Y leaves 4 days before the completion of the work. In how many days in total is the work completed?

Introduction / Context:
This is a slightly advanced time and work question involving three workers X, Y, and Z with different individual times. They start together, but Y leaves before the work ends. We need to handle two time intervals: a period when all three work together, and a final period when only X and Z work. This requires careful algebra and use of total work as 1 unit.

Given Data / Assumptions:
- X alone can finish the work in 30 days.
- Y alone can finish the work in 40 days.
- Z alone can finish the work in 50 days.
- All three start together.
- Y leaves 4 days before completion, so during the last 4 days only X and Z work.
- Total work is taken as 1 unit.

Concept / Approach:
Let T be the total number of days the work takes. Then for the first T - 4 days, all three work together. For the last 4 days, only X and Z work. We convert individual times into rates, write an equation for the total work done, and solve for T. Fractional arithmetic is essential because the answer is a rational number rather than a neat integer.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: Rate of X = 1 / 30 work per day. Step 3: Rate of Y = 1 / 40 work per day. Step 4: Rate of Z = 1 / 50 work per day. Step 5: Let T be the total time in days for completion. Step 6: For T - 4 days, X, Y, and Z work together. For the last 4 days, only X and Z work. Step 7: Total work done by X = T / 30 units. Step 8: Total work done by Y = (T - 4) / 40 units. Step 9: Total work done by Z = T / 50 units. Step 10: Since full work is 1 unit, we have T / 30 + (T - 4) / 40 + T / 50 = 1. Step 11: Solving this equation gives T = 660 / 47 days (approximately 14.04 days).

Verification / Alternative check:
We can verify numerically: T is about 14.04 days, so Y works for about 10.04 days and X and Z work for the full period. Using precise fractions, substitute T = 660 / 47 into T / 30, (T - 4) / 40, and T / 50, add the three values, and they sum to 1 unit of work. This confirms that T = 660 / 47 is correct.

Why Other Options Are Wrong:
- 640/47, 680/47, 665/47: These nearby fractions produce totals slightly less or more than the full unit of work when substituted into the equation, so they do not satisfy the work condition exactly.
- 14: This is an approximation but not the exact solution, and leads to a small imbalance when total work is recomputed.

Common Pitfalls:
Students often assume that all three work for the same duration or forget to adjust Y’s time by subtracting 4 days. Another common mistake is not writing a proper equation for total work done, or handling fractions incorrectly when solving. It is important to separate the time intervals clearly and carefully use algebraic methods to find T.

Final Answer:
The total time taken to complete the work is 660/47 days.