Work Calculation: Step-by-Step Solution

This problem requires calculating the total time needed to complete a painting task undertaken by three individuals, considering their varying work rates and changes in working patterns.

Individual Work Rates

First, we determine the work rate for each person. The rate signifies the fraction of the total work completed per hour.

• Ravina's time = 32 hours. Ravina's rate = $ \frac{1}{32} $ work/hour.
• Sujata's time = 48 hours. Sujata's rate = $ \frac{1}{48} $ work/hour.
• Saroj's time = 64 hours. Saroj's rate = $ \frac{1}{64} $ work/hour.

Work Done Together (First 5 Hours)

Ravina, Sujata, and Saroj begin the work together. To find their combined rate, we sum their individual rates.

Combined Rate = $ \frac{1}{32} + \frac{1}{48} + \frac{1}{64} $

The Least Common Multiple (LCM) of 32, 48, and 64 is 192. Converting the fractions:

Combined Rate = $ \frac{6}{192} + \frac{4}{192} + \frac{3}{192} = \frac{13}{192} $ work per hour.

The work done in the initial 5 hours is calculated as:

Work Done = Combined Rate $ \times $ Time = $ \frac{13}{192} \times 5 = \frac{65}{192} $.

Remaining Work Calculation

After 5 hours, Saroj departs. The total work is represented as 1 unit.

Remaining Work = 1 - Work Done = $ 1 - \frac{65}{192} = \frac{192 - 65}{192} = \frac{127}{192} $.

Work Done Alternatingly

From the 6th hour onwards, Ravina and Sujata work in turns, with Ravina starting. This pattern continues hour by hour.

For any 2-hour period where Ravina works for 1 hour and Sujata works for 1 hour:

Work done in 2 hours = Ravina's Rate + Sujata's Rate = $ \frac{1}{32} + \frac{1}{48} $

Using LCM = 96:

Work done in 2 hours = $ \frac{3}{96} + \frac{2}{96} = \frac{5}{96} $.

Let's consider the work done over the next 25 hours (from hour 6 up to hour 30).

In these 25 hours, Ravina works for $\lceil 25/2 \rceil = 13$ hours (hours 6, 8, ..., 30).

Sujata works for $\lfloor 25/2 \rfloor = 12$ hours (hours 7, 9, ..., 29).

Work done in these 25 hours = $ (13 \times \frac{1}{32}) + (12 \times \frac{1}{48}) $

= $ \frac{13}{32} + \frac{12}{48} = \frac{13}{32} + \frac{1}{4} = \frac{13}{32} + \frac{8}{32} = \frac{21}{32} $.

Converting this to the common denominator 192:

$ \frac{21}{32} = \frac{21 \times 6}{32 \times 6} = \frac{126}{192} $.

Total work done after 30 hours (5 hours together + 25 hours alternating) is:

Total Work = $ \frac{65}{192} + \frac{126}{192} = \frac{191}{192} $.

Final Time Calculation

At the end of 30 hours, the amount of work remaining is:

Remaining Work = $ 1 - \frac{191}{192} = \frac{1}{192} $.

This final fraction of work is completed in the subsequent 15 minutes.

Therefore, the total time taken to complete the painting work is the sum of the time spent in different phases:

Total Time = 5 hours (working together) + 25 hours (alternating phase) + 15 minutes (final completion)

Total Time = 30 hours + 15 minutes = 30 hours 15 minutes.