A can complete a certain work in 21 days and B can complete the same work in 42 days. If they work together on this work for 7 days and then stop, what fraction of the total work will still remain unfinished?

Introduction / Context:

This time and work problem asks for the fraction of work remaining after a certain period of joint work by two workers with different individual efficiencies. It checks the ability to convert individual completion times into daily rates, sum those rates for joint work, and then interpret the fraction completed versus the fraction remaining. Such questions are common in entrance exams and quantitative aptitude tests.

Given Data / Assumptions:

• A can complete the entire work alone in 21 days.
• B can complete the entire work alone in 42 days.
• A and B work together for 7 days.
• We assume the work is one unit and that rates are constant throughout.

Concept / Approach:

The method is to convert the time taken by each worker alone into a daily work rate. For example, if someone completes the work in 21 days, that person completes 1/21 of the work in one day. When two workers collaborate, their daily rates add. The total fraction of work completed in the given number of days is simply joint rate multiplied by number of days. Subtracting this completed fraction from one gives the fraction that remains unfinished.

Step-by-Step Solution:

Verification / Alternative Check:

If only A worked, in 7 days A would complete 7 * 1/21 = 1/3 of the work. If only B worked, in 7 days B would complete 7 * 1/42 = 1/6 of the work. Together, 1/3 + 1/6 = 1/2, which agrees with the earlier calculation. This cross check confirms that 1/2 of the work is done and therefore 1/2 remains incomplete after 7 days of combined work.

Why Other Options Are Wrong:

Options like 1/3 or 1/4 represent smaller remaining fractions, implying that more than half the work was completed in 7 days, which contradicts the exact calculation of 1/2 completed. The option 2/3 would mean only 1/3 was done, which is also not supported by the computed joint rate. Only 1/2 matches the arithmetic based on the individual completion times and joint work period.

Common Pitfalls:

Some students mistakenly add the days (21 and 42) instead of the rates, or treat 7 days as a simple fraction of one worker's time without accounting for both working together. Another frequent mistake is to stop after finding the fraction completed, rather than converting it to the fraction remaining as the question asks. Carefully distinguishing between completed and remaining portions is essential.

Final Answer:

The fraction of the total work that remains unfinished after 7 days of joint work is 1/2 of the work.