A can complete a piece of work in 16 days and B can complete the same work in 20 days. If they work together on this work for 8 days and then stop, what fraction of the total work will still remain to be completed?

Introduction / Context:

This problem deals with the fraction of work left after two workers with different efficiencies have worked together for a certain number of days. It tests the basic use of work rates and joint work in the context of time and work, a very common topic in competitive exams and aptitude tests.

Given Data / Assumptions:

• A alone finishes the work in 16 days.
• B alone finishes the same work in 20 days.
• A and B work together for 8 days.
• We assume constant work rates and take the whole work as one unit.

Concept / Approach:

The method is to convert the given completion times into daily work rates, add these to find the combined rate when A and B work together, and then compute the amount of work finished in the given number of days. The remaining fraction of work is obtained by subtracting the completed amount from the total work, which is taken as one.

Step-by-Step Solution:

Verification / Alternative Check:

Another way to check is to think in terms of equivalent days of one worker. In one day, A and B together do 9/80 of the work. To complete the full work at this rate would take 80/9 days, which is slightly more than 8 days. After 8 days they should therefore be close to finishing, and indeed 9/10 of the work is completed. The remaining 1/10 is consistent with this reasoning and matches the fractional calculation.

Why Other Options Are Wrong:

A remaining fraction of 1/3 or 1/6 would mean much less work has been completed than the known rates allow in 8 days. A remaining fraction of 2/9 would mean only 7/9 is complete, which does not match the product 8 * 9/80 = 9/10. Only 1/10 is consistent with the work rate calculations and the time given.

Common Pitfalls:

Students sometimes mistakenly average the days (16 and 20) or confuse them with the number of days worked. Another frequent error is to miscalculate the sum of 1/16 and 1/20 or neglect to simplify the resulting fraction properly. Keeping track of fractions and clearly distinguishing between rate and time helps avoid these mistakes.

Final Answer:

The fraction of the work that remains unfinished after 8 days of joint work is 1/10 of the total work.