A can complete a work in 36 days and B can complete the same work in 12 days. If they work together on the job for 3 days, what fraction of the work will still remain unfinished?

Introduction / Context:
This problem looks at partial completion when two workers A and B work together for only a few days. You are given how long each would take individually to complete the whole job. The question is to determine what fraction of the job remains after they have worked together for 3 days. This type of question is fundamental for understanding how rates combine and how to track progress over time.

Given Data / Assumptions:

• A alone can complete the work in 36 days.
• B alone can complete the work in 12 days.
• A and B work together for 3 days only.
• We assume constant work rates for both workers.
• We must find the fraction of the total work that remains unfinished after these 3 days.

Concept / Approach:
We treat the total work as 1 unit. A's rate and B's rate are each 1 divided by their respective total times. The combined rate is the sum of these two rates. Multiplying the combined rate by 3 (the number of days worked) gives the fraction of the job completed. Subtracting this from 1 yields the remaining fraction of the job. This is a direct application of work rate arithmetic.

Step-by-Step Solution:
Step 1: Let total work = 1 unit. Step 2: A's daily rate = 1/36 of the work per day. Step 3: B's daily rate = 1/12 of the work per day. Step 4: Combined rate of A and B = 1/36 + 1/12. Step 5: Use a common denominator of 36: 1/12 = 3/36, so combined rate = 1/36 + 3/36 = 4/36 = 1/9 of the work per day. Step 6: Work completed in 3 days together = 3 × (1/9) = 3/9 = 1/3 of the total work. Step 7: Remaining work = total work − work completed = 1 − 1/3 = 2/3 of the work.

Verification / Alternative check:
We can check by reasoning. If A and B keep working together at 1/9 of the job per day, they will need 9 days to finish the full job. In 3 days, they have done 3/9 of the job, which is 1/3. Therefore, 2/3 remains. This matches the result, and the full-time check also aligns, since 9 days is less than both 36 and 12 days, as expected for combined work.

Why Other Options Are Wrong:
1/3 would be the fraction completed, not the fraction remaining. 1/4, 1/5 and 3/4 do not match the arithmetic of 1 − 1/3. 3/4 actually exceeds the completed fraction of 1/3 and would correspond to having done much more work than the pair actually did in just 3 days. Therefore, the only correct fraction for the remaining work is 2/3.

Common Pitfalls:
One frequent mistake is to compute the fraction completed correctly and then incorrectly identify it as the remaining fraction. Another is to misadd the rates 1/36 and 1/12 without using a proper common denominator. Some also forget to multiply the combined rate by the number of days. Keeping the sequence "find rate, multiply by time, subtract from total" helps avoid these issues.

Final Answer:
After 3 days of working together, two-thirds (2/3) of the work still remains unfinished.