Problem Breakdown:

• The contractor initially engaged 54 workers to complete a job in 21 days.
• After working for 3 days, the contractor realizes that if he continues with the same number of workers, the job will take 28 days to complete.
• The task is to find out how many additional workers are needed to complete the job in the original 21 days timeframe.

Step-by-Step Solution:

1. Determine the total amount of work:

   Total work can be expressed in terms of worker-days (number of workers × number of days).

   Initially, the contractor planned to complete the job with 54 workers in 21 days. So, the total amount of work (W) is: $W = 54 \times 21 = 1134 \text{ worker-days}.$

2. Calculate the work done in the first 3 days:

   In the first 3 days, the contractor worked with 54 workers. Therefore, the amount of work completed in these 3 days is: $\text{Work completed in 3 days} = 54 \times 3 = 162 \text{ worker-days}.$

3. Calculate the remaining work:

   The remaining work after 3 days is the total work minus the work completed in these 3 days: $\text{Remaining work} = 1134 - 162 = 972 \text{ worker-days}.$

4. Determine the new rate of work to finish in 28 days:

   The contractor realizes that with the same 54 workers, the remaining work would take 28 days in total. So, the number of worker-days needed for the remaining work if 54 workers are employed for 28 days: $\text{Work done by 54 workers in 28 days} = 54 \times 28 = 1512 \text{ worker-days}.$ However, the remaining work is only 972 worker-days, so this is why the timeline increases to 28 days.

5. Calculate how many workers are needed to finish in the original 21 days:

   The contractor wants to complete the job in the original 21 days. After 3 days of work, there are 18 days left. Let the number of workers needed to complete the remaining work in 18 days be x.

   The total work remaining (972 worker-days) should now be completed by these x workers in 18 days: $\text{Remaining work} = x \times 18.$ Equating this to the remaining work: $x \times 18 = 972.$ Solving for x: $x = \frac{972}{18} = 54 \text{ workers}.$

   So, the contractor needs 54 workers to complete the remaining work in 18 days.

6. How many more workers to engage?

   The contractor already has 54 workers. Therefore, the number of additional workers needed is: $\text{Additional workers} = 54 - 54 = 0.$

   Therefore, no additional workers are needed to complete the job in the remaining time.

Conclusion:

No additional workers are needed. The current workforce of 54 workers can complete the job in 21 days.

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Do you have any questions, or would you like more details on a particular part of the solution?

Here are 5 additional questions to expand on this:

1. How would the solution change if the contractor had less than 54 workers initially?
2. What if the contractor wanted to finish the job in fewer than 21 days?
3. How do worker efficiency and productivity affect these types of problems?
4. Can you determine the amount of work done per day by each worker?
5. How would delays in the work affect the number of workers needed?

Tip: In time-based work problems, keeping track of the amount of work done per day is key to solving efficiently.