Solving the Work and Time Problem with Alternate Days

This problem involves two individuals, Rashmi and Tina, working on a task on alternate days. Rashmi starts the work. We need to find the total number of days required to complete the entire work.

Understanding Individual Work Rates

First, let's determine how much work each person can do in a single day. This is their work rate.

• Rashmi can do the work in 20 days. So, Rashmi's daily work rate is \( \frac{1}{20} \) of the total work per day.
• Tina can do the work in 30 days. So, Tina's daily work rate is \( \frac{1}{30} \) of the total work per day.

Work Done in One Cycle (Two Days)

Since they work on alternate days with Rashmi starting, the work happens in cycles of two days:

• Day 1: Rashmi works.
• Day 2: Tina works.

Let's calculate the total amount of work done in one such two-day cycle.

Work done in 1 cycle = Work done by Rashmi on Day 1 + Work done by Tina on Day 2

Work done in 1 cycle = \( \frac{1}{20} + \frac{1}{30} \)

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

\( \frac{1}{20} = \frac{1 \times 3}{20 \times 3} = \frac{3}{60} \)

\( \frac{1}{30} = \frac{1 \times 2}{30 \times 2} = \frac{2}{60} \)

Work done in 1 cycle = \( \frac{3}{60} + \frac{2}{60} = \frac{5}{60} = \frac{1}{12} \)

So, in every two-day cycle, \( \frac{1}{12} \) of the total work is completed.

Calculating the Number of Cycles

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

If 1 cycle completes \( \frac{1}{12} \) of the work, then to complete 1 whole work, the number of cycles needed is:

Number of cycles = Total work ÷ Work done in 1 cycle

Number of cycles = \( 1 \div \frac{1}{12} = 1 \times \frac{12}{1} = 12 \)

So, exactly 12 cycles are needed to complete the work.

Total Days to Complete the Work

Each cycle consists of 2 days. Since 12 cycles are needed, the total number of days will be:

Total days = Number of cycles × Days per cycle

Total days = \( 12 \times 2 = 24 \) days

Let's quickly verify this:

• After 11 cycles (22 days), \( 11 \times \frac{1}{12} = \frac{11}{12} \) of the work is done.
• Remaining work = \( 1 - \frac{11}{12} = \frac{1}{12} \).
• Day 23 is the start of the 12th cycle, which is Rashmi's turn. Rashmi does \( \frac{1}{20} \) work.
• Work remaining after Day 23 = \( \frac{1}{12} - \frac{1}{20} \). Common denominator 60: \( \frac{5}{60} - \frac{3}{60} = \frac{2}{60} = \frac{1}{30} \).
• Day 24 is Tina's turn. Tina's daily work rate is \( \frac{1}{30} \).
• Tina completes the remaining \( \frac{1}{30} \) work on Day 24.
• The work is finished exactly at the end of Day 24.

Therefore, the total number of days required to complete the work is 24 days.

Conclusion

By calculating the work done in a two-day cycle and determining the number of cycles needed, we find that the work is completed in 24 days when Rashmi and Tina work on alternate days starting with Rashmi.

Revision Table: Alternate Day Work Problems

Here's a quick review of the steps for alternate day work problems:

Additional Information: Work and Time Concepts

Work and time problems often involve calculating efficiency and combining work rates.

• Efficiency: A person's efficiency is inversely proportional to the time taken to complete a task. More efficient people take less time.
• Combined Work Rate: If two people A and B can do a work in \( t_A \) and \( t_B \) days respectively, their combined daily work rate when working together is \( \frac{1}{t_A} + \frac{1}{t_B} \). The total time taken together is \( \frac{1}{\frac{1}{t_A} + \frac{1}{t_B}} = \frac{t_A \times t_B}{t_A + t_B} \).
• Alternate Days: As seen in this problem, the work is not continuous but happens in a sequence, requiring calculation based on cycles.

Understanding these basic concepts helps in solving various types of work and time problems.