Understanding the Work and Time Problem

This question involves the concept of Work and Time. We are given the time taken by three individuals, 'A', 'B', and 'C', to complete a piece of work independently. We need to find the total time taken to finish the work when 'A' works consistently, and 'B' and 'C' join him on every third day.

Calculating Individual Daily Work Rates

First, let's determine the amount of work each person can complete in one day. The total work is considered as 1 unit.

• 'A' can do the work in 20 days. So, A's daily work rate is \( \frac{1}{20} \).
• 'B' can do the work in 30 days. So, B's daily work rate is \( \frac{1}{30} \).
• 'C' can do the work in 60 days. So, C's daily work rate is \( \frac{1}{60} \).

Analyzing the Work Pattern

The work pattern is as follows:

• Day 1: 'A' works alone.
• Day 2: 'A' works alone.
• Day 3: 'A', 'B', and 'C' work together.

This cycle of 3 days repeats until the work is completed.

Work Done in One Cycle (3 Days)

Let's calculate the total work done in one 3-day cycle:

• Work done on Day 1 by A = \( \frac{1}{20} \)
• Work done on Day 2 by A = \( \frac{1}{20} \)
• Work done on Day 3 by A, B, and C together = A's daily rate + B's daily rate + C's daily rate
• Combined daily rate of A, B, and C = \( \frac{1}{20} + \frac{1}{30} + \frac{1}{60} \)

To add these fractions, we find a common denominator, which is the Least Common Multiple (LCM) of 20, 30, and 60. The LCM is 60.

Combined daily rate = \( \frac{3}{60} + \frac{2}{60} + \frac{1}{60} = \frac{3+2+1}{60} = \frac{6}{60} = \frac{1}{10} \)

So, the work done on Day 3 by A, B, and C together is \( \frac{1}{10} \).

Total work done in one 3-day cycle = (Work on Day 1) + (Work on Day 2) + (Work on Day 3)

Total work in 3 days = \( \frac{1}{20} + \frac{1}{20} + \frac{1}{10} = \frac{2}{20} + \frac{1}{10} = \frac{1}{10} + \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \)

Thus, \( \frac{1}{5} \) of the work is completed in every 3-day cycle.

Calculating the Number of Cycles

To complete the entire work (which is 1 unit), we need to find out how many such 3-day cycles are required.

Number of cycles = \( \frac{\text{Total Work}}{\text{Work Done in one Cycle}} = \frac{1}{\frac{1}{5}} = 1 \times 5 = 5 \) cycles.

Calculating Total Days

Since each cycle takes 3 days, the total number of days to complete the work is:

Total days = Number of cycles \( \times \) Days per cycle

Total days = \( 5 \times 3 = 15 \) days.

Conclusion

Following the pattern where 'A' works alone for two days and is assisted by 'B' and 'C' on every third day, the total work will be completed in 15 days.