A contractor undertook a project that would normally take 20 days with 5 workers working continuously. The client asked for an earlier completion, so the contractor decided to add 5 additional workers every 2 days, starting from 5 workers on day one, in order to finish in the new time. Later, the total work was increased by 50 percent, but the contractor continued to add 5 workers every 2 days in the same pattern. How many more days, beyond the client's revised completion time, will now be required to finish the increased work?

Introduction / Context:
This is a challenging time and work question involving a changing workforce and an increase in total work. The contractor initially plans to complete the work in 20 days with 5 workers. To satisfy the client, he speeds up progress by adding more workers at fixed intervals. After that, the total amount of work is increased by 50 percent, and the same staffing pattern continues. We must determine the extra time required beyond the client's new deadline. This type of problem tests both arithmetic series reasoning and work rate concepts.

Given Data / Assumptions:

• Original plan: 5 workers for 20 days, so original work equals 100 worker days.
• To finish earlier, he starts with 5 workers and increases by 5 every 2 days.
• The client's revised time is the new completion time with this increasing workforce pattern for the original work.
• Later, the total work increases by 50 percent, becoming 150 worker days.
• The contractor still increases the workforce by 5 every 2 days in the same way.
• We assume each worker has constant efficiency and that increases occur at the beginning of each 2 day block.

Concept / Approach:
The total work can be viewed in worker days. With the increasing workforce, each 2 day period has a fixed number of workers. We can model the workforce over 2 day blocks as 5, 10, 15, 20, and so on. The total worker days in n such blocks form an arithmetic series in terms of man days. First we calculate how many blocks are needed to reach the original 100 worker days and hence determine the client's revised completion time. Then we extend the same pattern to reach 150 worker days and compare this new time with the earlier completion time to find how many extra days are needed.

Step-by-Step Solution:
Step 1: Original work is 5 workers times 20 days = 100 worker days. Step 2: With the new pattern, workers in each 2 day block are 5, 10, 15, and 20. Let there be n blocks. Workers in the i-th block equal 5i. Step 3: Worker days over n blocks equal 2 times the sum from i = 1 to n of 5i, which is 10 times n(n + 1) / 2 = 5n(n + 1). Step 4: For the original 100 worker days, set 5n(n + 1) = 100, giving n(n + 1) = 20 and n = 4. So the client's revised completion time is 4 blocks times 2 days each = 8 days. Step 5: Total work is increased by 50 percent, so new required worker days are 1.5 times 100 = 150 worker days. Step 6: By 8 days (4 blocks), 100 worker days have been done, leaving 50 worker days. Step 7: The next block from day 8 to day 10 has 25 workers for 2 days, supplying exactly 25 times 2 = 50 worker days. So the increased work is completed at day 10.

Verification / Alternative check:
We can list the worker days explicitly. Days 1 to 2 use 5 workers, giving 10 worker days. Days 3 to 4 use 10 workers, giving 20 worker days, cumulative 30. Days 5 to 6 use 15 workers, giving 30 more, cumulative 60. Days 7 to 8 use 20 workers, giving 40 more, cumulative 100. For the extra 50 worker days, days 9 to 10 use 25 workers, giving 50 worker days, bringing the total to 150. This matches the 50 percent increase over the original 100 worker days and confirms the 10 day total duration.

Why Other Options Are Wrong:

• 1 day more would give only half of the additional 50 worker days at most, so the work would not be finished.
• 3 days or 5 days more than 8 days would overshoot the required 150 worker days by a large margin.
• Only 2 extra days beyond the 8 days give exactly the additional 50 worker days needed.

Common Pitfalls:
Many learners misinterpret the phrase increase 5 workers every 2 days and mix up the arithmetic progression of workers with the arithmetic of worker days. Another common mistake is to recompute from scratch for the increased work instead of using the earlier total for the original work. Treating work consistently in worker days and carefully summing the arithmetic series over each block avoids these errors.

Final Answer:
With the increased work and the continuing pattern of adding workers, an additional 2 days beyond the client's revised completion time are required, so the total time becomes 10 days.