Construction Project Time Calculation

This problem involves calculating the total duration of a construction project where the number of workers and their efficiency change over time. We need to break down the project into phases based on these changes and calculate the work done in each phase.

Understanding the Total Work Required

First, let's determine the total amount of work required for the project. We assume the initial 'work' unit is equivalent to one worker working for one day at the original efficiency level.

• Number of original workers = 18
• Number of days planned = 25
• Total work = Number of workers × Number of days
• Total work = $18 \times 25 = 450$ worker-days (at original efficiency).

Phase 1: Initial Work

The project starts with 18 workers who work for the first 10 days.

• Workers = 18
• Days worked = 10
• Work done in Phase 1 = $18 \text{ workers} \times 10 \text{ days} = 180$ worker-days.
• Remaining work = Total work - Work done in Phase 1
• Remaining work = $450 - 180 = 270$ worker-days.

Phase 2: Changes in Workforce

After 10 days, 6 workers leave, and 4 new workers join. These new workers are 25% more efficient than the original workers.

• Workers remaining after 6 leave = $18 - 6 = 12$ original workers.
• New workers joining = 4.
• Total number of workers = $12 + 4 = 16$ workers.
• Efficiency of new workers = $100\% + 25\% = 125\%$ of original efficiency.

Let the original efficiency of a worker be $E$. The efficiency of the 12 original workers is $12 \times E$. The efficiency of the 4 new workers is $4 \times (1.25 \times E) = 5 \times E$.

The combined effective efficiency of the workforce in this phase is $(12 \times E) + (5 \times E) = 17 \times E$. This means the group works at a rate equivalent to 17 original workers.

This group works for the next 5 days.

• Days worked in Phase 2 = 5
• Work done in Phase 2 = (Effective efficiency) × Days worked
• Work done in Phase 2 = $(17 \times E) \times 5 \text{ days} = 85 \times E$ worker-days.
• Remaining work after Phase 2 = Work remaining after Phase 1 - Work done in Phase 2
• Remaining work = $270 - 85 = 185$ worker-days.

Phase 3: Efficiency Increase

After these 5 days (i.e., at the start of Phase 3), all remaining workers increase their efficiency by 20%.

• Total workers = 16 (12 original + 4 new).
• Efficiency increase = 20%.

Let's calculate the new efficiencies:

• The 12 original workers (initial efficiency $E$) now work at $E \times (1 + 0.20) = 1.20E$.
• The 4 new workers (initial efficiency $1.25E$) now work at $(1.25E) \times (1 + 0.20) = 1.25E \times 1.20 = 1.50E$.

The new combined effective efficiency is:

• Effective efficiency = $(12 \times 1.20E) + (4 \times 1.50E)$
• Effective efficiency = $14.4E + 6.0E = 20.4E$.

This means the current group of 16 workers works at a rate equivalent to 20.4 original workers.

Calculating Remaining Days

We need to find out how many days are required to complete the remaining 185 worker-days of work with this new efficiency level.

• Remaining work = 185 worker-days.
• Effective rate of work = $20.4$ original worker-days per day.
• Days needed for Phase 3 = $\frac{\text{Remaining work}}{\text{Effective rate of work}}$
• Days needed for Phase 3 = $\frac{185}{20.4}$ days.

To simplify the fraction:

• $\frac{185}{20.4} = \frac{1850}{204}$
• Divide both numerator and denominator by 2: $\frac{925}{102}$ days.

Convert the improper fraction to a mixed number:

• $925 \div 102 = 9$ with a remainder of $7$ ($102 \times 9 = 918$).
• So, Days needed for Phase 3 = $9\frac{7}{102}$ days.

Total Project Duration

The total time taken for the project is the sum of the days spent in each phase.

• Total Days = Days in Phase 1 + Days in Phase 2 + Days in Phase 3
• Total Days = $10 + 5 + 9\frac{7}{102}$
• Total Days = $24\frac{7}{102}$ days.

Therefore, the project will take a total of $24\frac{7}{102}$ days to complete.