A can finish a piece of work in 18 days and B can finish the same work in 36 days. If they work together on the job for 9 days, what percentage of the work is still left unfinished?

Introduction / Context:
This question is about determining how much work remains after two workers, A and B, have worked together for a limited period. We are given the individual times for A and B to complete the job alone. The task is to find the percentage of work left after they jointly work for 9 days. This tests understanding of combined work rates and conversion of a fraction of work into a percentage remaining.

Given Data / Assumptions:
- A alone can complete the work in 18 days. - B alone can complete the same work in 36 days. - A and B work together for 9 days. - Work rates remain constant throughout. - Total work is one full job.

Concept / Approach:
The daily work rate of a person who takes T days to finish a job is 1 / T of the job per day. When two people work together, we add their rates to get the combined rate. We multiply this combined rate by the number of working days to find the fraction of the job completed. Subtracting this from 1 gives the fraction remaining. Finally, we convert that fraction into a percentage to answer the question.

Step-by-Step Solution:
Step 1: A's daily work rate = 1 / 18 of the job per day. Step 2: B's daily work rate = 1 / 36 of the job per day. Step 3: Combined daily rate = 1 / 18 + 1 / 36. Step 4: Take a common denominator 36: 1 / 18 = 2 / 36, so combined rate = 2 / 36 + 1 / 36 = 3 / 36 = 1 / 12 job per day. Step 5: Work done in 9 days at this rate = 9 * (1 / 12) = 9 / 12 = 3 / 4 of the job. Step 6: Fraction of work remaining = 1 - 3 / 4 = 1 / 4. Step 7: Convert 1 / 4 to percentage: (1 / 4) * 100 percent = 25 percent.

Verification / Alternative check:
We can verify by checking how long it would take them to finish the job together. At 1 / 12 job per day, they need 12 days to finish the entire job. Working for 9 days completes 9 / 12 = 3 / 4 of the job, leaving exactly 1 / 4. Since 1 / 4 of 12 days is 3 days, they would need 3 more days to finish, which is consistent with the idea that 25 percent of the work remains.

Why Other Options Are Wrong:
33.3 or 50: These represent 1 / 3 and 1 / 2 of the job remaining, which do not match the combined rate and time used.
20: This corresponds to 1 / 5 of the job remaining, not 1 / 4.
75: This would mean only 25 percent completed and 75 percent left, which contradicts the calculated 3 / 4 already completed.

Common Pitfalls:
A common mistake is to miscalculate the combined rate by averaging times instead of adding rates. Another error is to misinterpret the question and give the percentage of work completed instead of the percentage remaining. Always read the question carefully to see whether it asks for the part done or the part left, and consistently convert fractions to percentages.

Final Answer:
The percentage of work left unfinished after 9 days is 25 percent.