P alone can complete a piece of work in 6 days. The work done by Q alone in one day is equal to one third of the work done by P alone in one day. If P and Q work together from the beginning, in how many days will they complete the work?

Introduction / Context:
This is a fundamental time and work problem with two workers having related efficiencies. We know how long P takes alone and how Q daily work rate compares to that of P. The task is to determine the time needed when both work together from the start.

Given Data / Assumptions:
• P alone completes the work in 6 days.
• Q daily work is one third of P daily work.
• Both P and Q start working together and continue until completion.
• Work rates remain constant.

Concept / Approach:
We translate the given times and relationships into daily work fractions. P daily rate is the reciprocal of his time. Q daily rate is then defined as one third of that. Adding the two daily rates gives a combined daily work fraction. The reciprocal of this combined rate yields the total number of days required to finish the work together.

Step-by-Step Solution:
Assume total work = 1 job. P daily work rate = 1 / 6 of the job per day. Q daily work rate is one third of P rate, so Q rate = (1 / 3) × (1 / 6) = 1 / 18 of the job per day. Combined daily work rate of P and Q = 1 / 6 + 1 / 18. Find a common denominator 18: 1 / 6 = 3 / 18, so total = 3 / 18 + 1 / 18 = 4 / 18. Simplify: 4 / 18 = 2 / 9 of the job per day. Time required to finish the job together = 1 ÷ (2 / 9) = 9 / 2 days. As a mixed number, 9 / 2 days = 4 1/2 days.

Verification / Alternative check:
We can verify by computing total work done in 4 1/2 days. P contribution = 4.5 × 1 / 6 = 4.5 / 6 = 0.75 of the job. Q contribution = 4.5 × 1 / 18 = 4.5 / 18 = 0.25 of the job. Total work = 0.75 + 0.25 = 1 complete job, confirming the calculation.

Why Other Options Are Wrong:
Times longer than 6 days are impossible because P alone finishes in 6 days, so adding Q cannot increase the total time. Times such as 5 2/3 or 6 3/4 days do not match the combined rate of 2 / 9 job per day and would lead to too much or too little work done compared with the total job.

Common Pitfalls:
Some learners misinterpret “one third of the work done by P” as Q finishing the work in three times the total time of P rather than one third of P daily rate. This confusion changes the relation between rates and leads to errors. Always define rates per day and then compare them carefully.

Final Answer:
P and Q together will complete the work in 4 1/2 days.