Team A and Team B Work Rates

We start by calculating the individual work rates of Team A and Team B, assuming the total job is 1 unit of work.

• Team A's Rate ($R_A$): Completes the job in 15 days, so $R_A = \frac{1}{15}$ job/day.
• Team B's Rate ($R_B$): Completes the job in 20 days, so $R_B = \frac{1}{20}$ job/day.

Work Calculation Phase 1: Joint Work

Teams A and B work together for 6 days. Their combined rate is:

$R_{A+B} = R_A + R_B = \frac{1}{15} + \frac{1}{20}$

To add these, we find a common denominator, which is 60:

$R_{A+B} = \frac{4}{60} + \frac{3}{60} = \frac{7}{60}$ job/day.

Work done in these 6 days ($W_1$) is calculated as:

$W_1 = \text{Combined Rate} \times \text{Time} = \frac{7}{60} \times 6 = \frac{7}{10}$ job.

Work Calculation Phase 2: Team A Alone

Team A works alone for the next 3 days. The work done during this phase ($W_2$) is:

$W_2 = R_A \times \text{Time} = \frac{1}{15} \times 3 = \frac{3}{15} = \frac{1}{5}$ job.

Total Work Done and Remaining Job

The total amount of work completed after these two phases is the sum of $W_1$ and $W_2$:

Total Work Done = $W_1 + W_2 = \frac{7}{10} + \frac{1}{5}$

Using a common denominator of 10:

Total Work Done = $\frac{7}{10} + \frac{2}{10} = \frac{9}{10}$ job.

The remaining work required to finish the job is:

Remaining Work = Total Job - Total Work Done = $1 - \frac{9}{10} = \frac{1}{10}$ job.

Efficiency Changes for Phase 3

In the final phase, the teams' work efficiencies change:

• Team A's efficiency reduces to 60% of its original rate. The new rate ($R_{A\_new}$) is:

  $R_{A\_new} = 0.60 \times R_A = \frac{6}{10} \times \frac{1}{15} = \frac{3}{5} \times \frac{1}{15} = \frac{1}{5} \times \frac{1}{5} = \frac{1}{25}$ job/day.

• Team B's efficiency increases to 120% of its original rate. The new rate ($R_{B\_new}$) is:

  $R_{B\_new} = 1.20 \times R_B = \frac{12}{10} \times \frac{1}{20} = \frac{6}{5} \times \frac{1}{20} = \frac{3}{5} \times \frac{1}{10} = \frac{3}{50}$ job/day.

Days Calculation for Remaining Job

Now, both teams work together again with their modified rates. Their combined rate for this phase ($R_{Phase3}$) is:

$R_{Phase3} = R_{A\_new} + R_{B\_new} = \frac{1}{25} + \frac{3}{50}$

Using a common denominator of 50:

$R_{Phase3} = \frac{2}{50} + \frac{3}{50} = \frac{5}{50} = \frac{1}{10}$ job/day.

The time needed to complete the remaining $\frac{1}{10}$ job is calculated by dividing the remaining work by the combined rate:

Time = $\frac{\text{Remaining Work}}{R_{Phase3}} = \frac{1/10 \text{ job}}{1/10 \text{ job/day}} = 1$ day.

Therefore, it will take exactly 1 more day to complete the job.