A shopkeeper must print a fixed number of documents and has three machines P, Q and R for this job. Machine P can complete the entire job in 3 days, machine Q in 4 days and machine R in 6 days. If all three machines are used simultaneously from the start, in how many days will the printing job be completed?

Introduction / Context:
This question focuses on combined work rates when multiple machines are working together. Machines P, Q and R can each complete the entire job individually in different amounts of time. We are required to find the total time taken when all three operate simultaneously. Such problems are very common in aptitude tests in the Time and Work chapter, especially in scenarios involving machines, taps or workers.

Given Data / Assumptions:

• Machine P completes the job in 3 days.
• Machine Q completes the job in 4 days.
• Machine R completes the job in 6 days.
• All three work together from the beginning.
• Total work is considered as 1 complete job.
• Each machine works at a constant rate.

Concept / Approach:
We convert the individual times into daily work rates. If a machine finishes a job in T days, its rate is 1 / T of the job per day. The combined rate is the sum of individual rates since they work simultaneously. Once we know the combined daily rate, the total time taken is the reciprocal of that combined rate, because time = work / rate and total work is 1 unit.

Step-by-Step Solution:
Let total work W = 1 unit (one complete printing job). Rate of P = 1 / 3 job per day. Rate of Q = 1 / 4 job per day. Rate of R = 1 / 6 job per day. Combined rate when P, Q and R work together = 1 / 3 + 1 / 4 + 1 / 6. Take LCM of 3, 4, 6 which is 12. 1 / 3 = 4 / 12, 1 / 4 = 3 / 12, 1 / 6 = 2 / 12. Combined rate = (4 + 3 + 2) / 12 = 9 / 12 = 3 / 4 job per day. Time taken when all three work together = total work / combined rate = 1 / (3 / 4) = 4 / 3 days.

Verification / Alternative check:
In 4 / 3 days, the total work done should equal 1 job. Check: (3 / 4) * (4 / 3) = 1, so the calculation is consistent. Alternatively, you can imagine 12 days as a convenient common multiple: in 12 days, P does 4 jobs, Q does 3 jobs and R does 2 jobs, totaling 9 jobs. Therefore, 1 job takes 12 / 9 = 4 / 3 days.

Why Other Options Are Wrong:
Option B (2 days) would imply a combined rate of 1 / 2 job per day, which is slower than P alone, so it cannot be correct. Option C (3/2 days) corresponds to a combined rate of 2 / 3 job per day, which does not match 1 / 3 + 1 / 4 + 1 / 6. Option D (4 days) is even slower and clearly not possible since each machine alone can finish in 3 to 6 days, and working together must be faster than the fastest individual time.

Common Pitfalls:
Many students make mistakes while adding fractions with different denominators. It is important to find the correct LCM before converting each rate into a common denominator. Another pitfall is to compute some sort of average of the times (3, 4 and 6 days) instead of using rates; averaging times does not work in work-rate problems and leads to incorrect answers.

Final Answer:
When machines P, Q and R all work together from the start, the printing job will be completed in 4/3 days (that is, 1 day and 8 hours).