A can complete a piece of work alone in 15 days and B can complete the same work alone in 9 days. If A and B work together on the job for 5 consecutive days, what fraction of the total work will still remain unfinished at the end?

Introduction / Context:
This question is about combining work rates of two people and then finding how much work remains after they have worked together for a certain period. It reinforces ideas about fractions of work completed and the relationship between time, rate and work in time and work problems.

Given Data / Assumptions:

A alone can finish the work in 15 days.
B alone can finish the same work in 9 days.
A and B work together for 5 days continuously at constant efficiencies.
The total work is considered as one complete job (1 unit).

Concept / Approach:
If someone completes a job in T days, the person's daily rate is 1/T. When two people work together, their combined rate is the sum of their individual rates. After calculating how much fraction of the work A and B perform in the 5 days, we subtract that from 1 to find the remaining fraction of the job that is still incomplete.

Step-by-Step Solution:
Let total work = 1 unit. Daily work rate of A = 1/15 of the work per day. Daily work rate of B = 1/9 of the work per day. Combined daily rate of A and B together = 1/15 + 1/9. Take LCM of 15 and 9, which is 45. Convert fractions: 1/15 = 3/45 and 1/9 = 5/45. Combined rate = 3/45 + 5/45 = 8/45 of the work per day. In 5 days, work completed = 5 * (8/45) = 40/45. Simplify 40/45 by dividing numerator and denominator by 5, giving 8/9 of the work completed. Remaining work = 1 - 8/9 = 1/9 of the total work.

Verification / Alternative check:
As a check, if 8/9 of the work is done in 5 days, then the average rate of the team is (8/9)/5 = 8/45 per day, which matches our calculated combined rate. Also, since A alone needs 15 days and B alone needs 9 days, it is reasonable that together they almost finish the work in 5 days but leave a small fraction. The remaining 1/9 is consistent with these intuitive expectations.

Why Other Options Are Wrong:
Option 1/6 suggests less work has been done, meaning more is left, but our calculation shows a higher combined rate and more work completed.
Option 3/7 does not come from any correct manipulation of the fractions and is not a natural result of the combined rate 8/45 over 5 days.
Option 1/3 would mean only two-thirds has been done, which is too little given the strong combined efficiency of A and B.
Option 2/9 is also incorrect because it corresponds to a completed fraction of 7/9, not 8/9, and does not match the exact multiplication of rate and time.

Common Pitfalls:
A common error is to average the times (15 and 9) instead of summing the work rates. Another frequent mistake is to miscalculate the LCM or the fractions, leading to an incorrect combined rate. Always convert to a common denominator and work carefully with fractions. Also, do not forget to subtract the completed fraction from 1 to find what is left.

Final Answer:
Therefore, the fraction of the work still left unfinished after 5 days is 1/9, so the correct option is 1/9.