Workers A, B, and C can complete a piece of work in 72, 48, and 36 days respectively when each works alone. For the first p/2 days, only A and B work together. For the next (p + 6)/3 days, A, B, and C all work together. The remaining 125/3 percent of the work is completed by worker D alone in 10 days. If C and D had instead worked together for p full days, what fraction of the total work would still remain unfinished?

Introduction / Context:
This is an advanced time and work problem involving four workers A, B, C, and D and a parameter p. Different people work during different intervals, and a certain percentage of the work is left for D. We must first determine the value of p from the given conditions, then imagine a new scenario where C and D work together for p days and compute what fraction of the work will still remain.

Given Data / Assumptions:

• A alone finishes the work in 72 days.
• B alone finishes in 48 days.
• C alone finishes in 36 days.
• For the first p/2 days, A and B work together.
• For the next (p + 6)/3 days, A, B, and C work together.
• The remaining 125/3 percent of the work is completed by D alone in 10 days.
• We interpret 125/3 percent as (125 / 3) / 100 = 5 / 12 of the total work.
• All rates are constant and work is measured as a single unit.

Concept / Approach:
We first convert the individual times into daily work rates. The statement that D completes the remaining 5 / 12 of the work in 10 days allows us to compute D's rate. Since that remaining fraction is the part left after A, B, and C have worked during the first two phases, we know that A, B, and C together complete 7 / 12 of the work in total during those phases. Setting up an equation for this 7 / 12 in terms of p gives us the value of p. Then, in the hypothetical scenario where C and D work together for p days, we use their combined rate to compute the fraction of work completed, and subtract this from 1 to find what fraction remains.

Step-by-Step Solution:
Rates: A's rate = 1 / 72, B's rate = 1 / 48, C's rate = 1 / 36. Let D's rate be d jobs per day. Remaining work fraction = 125 / 3 percent = 5 / 12. D completes 5 / 12 of the work in 10 days, so d * 10 = 5 / 12, hence d = (5 / 12) / 10 = 1 / 24. Total work done by A, B, and C during the first two phases is 1 - 5 / 12 = 7 / 12. Work in phase 1 (p/2 days by A and B) = (1 / 72 + 1 / 48) * (p / 2). Compute A + B: 1 / 72 + 1 / 48 = (2 / 144 + 3 / 144) = 5 / 144. So phase 1 work = (5 / 144) * (p / 2) = 5p / 288. Work in phase 2 ((p + 6)/3 days by A, B, and C) = (1 / 72 + 1 / 48 + 1 / 36) * ((p + 6) / 3). A + B + C = 1 / 72 + 1 / 48 + 1 / 36 = 1 / 16 (after simplifying). So phase 2 work = (1 / 16) * ((p + 6) / 3) = (p + 6) / 48. Total work by A, B, and C = 5p / 288 + (p + 6) / 48. Set this equal to 7 / 12 and solve for p: 5p / 288 + (p + 6) / 48 = 7 / 12. Converting to a common denominator and simplifying gives p = 12 days. Now consider the hypothetical scenario: C and D work together for p days = 12 days. C's rate = 1 / 36, D's rate = 1 / 24, so combined rate = 1 / 36 + 1 / 24 = (2 / 72 + 3 / 72) = 5 / 72 jobs per day. In 12 days, work done by C and D = 12 * (5 / 72) = 60 / 72 = 5 / 6. Remaining work fraction = 1 - 5 / 6 = 1 / 6 of the job.

Verification / Alternative check:
You can double check the computation of p by substituting p = 12 into the expressions for phase 1 and phase 2 work: phase 1 gives 5 * 12 / 288 = 60 / 288 = 5 / 24, while phase 2 gives (12 + 6) / 48 = 18 / 48 = 3 / 8. Adding 5 / 24 and 3 / 8 gives (5 / 24 + 9 / 24) = 14 / 24 = 7 / 12, confirming that 5 / 12 of the work is left for D. The second scenario with C and D also checks out as 5 / 6 completed and 1 / 6 remaining.

Why Other Options Are Wrong:
1/5, 1/7, and 1/8 would correspond to different values of p or different rates and do not match the carefully derived fraction. 1/9 is significantly smaller and would mean that C and D together complete a much larger fraction than they actually do according to their rates.

Common Pitfalls:
There are several places to go wrong: misreading 125/3 percent, forgetting that it represents 5 / 12 of the work; making algebraic mistakes when solving for p; or miscomputing the combined rates of A, B, C, and D. A clear step by step approach, careful fraction arithmetic, and checking intermediate results are essential for such multi stage problems.

Final Answer:
If C and D work together for p days, the fraction of work that will remain is 1/6 of the total.