Solving Work and Time Problems: Men Leaving the Job

This problem deals with the concept of work and time, specifically how the number of workers affects the time taken to complete a job. The core idea is that the total amount of work is constant, and it can be measured in 'man-days'. One man working for one day completes one unit of 'man-day' of work.

Calculating Total Work Units

We are given that 30 men can complete a job in 40 days. The total work required to complete the job can be calculated as the product of the number of men and the number of days they take.

Total Work = Number of Men × Number of Days

Total Work = \(30 \text{ men} \times 40 \text{ days} = 1200 \text{ man-days}\)

So, the entire job requires 1200 man-days of effort.

Work Done Before Men Left

The problem states that after 24 days, some men left. In the first 24 days, the original 30 men were working. We can calculate the amount of work completed during this period.

Work done in first 24 days = Number of Men × Number of Days worked

Work done in first 24 days = \(30 \text{ men} \times 24 \text{ days} = 720 \text{ man-days}\)

Calculating Remaining Work

Now we need to find out how much work is left to be done. This is the total work minus the work already completed.

Remaining Work = Total Work - Work done in first 24 days

Remaining Work = \(1200 \text{ man-days} - 720 \text{ man-days} = 480 \text{ man-days}\)

So, 480 man-days of work still need to be completed.

Finding the Number of Remaining Men

The remaining 480 man-days of work were completed by the remaining men in another 40 days. Let the number of men who remained on the job be \(M_{rem}\).

Remaining Work = Number of Remaining Men × Additional Days taken

\(480 \text{ man-days} = M_{rem} \text{ men} \times 40 \text{ days}\)

We can solve this equation to find \(M_{rem}\):

\(M_{rem} = \frac{480 \text{ man-days}}{40 \text{ days}}\)

\(M_{rem} = 12 \text{ men}\)

So, 12 men remained to complete the rest of the job.

Determining the Number of Men Who Left

Initially, there were 30 men on the job. After 24 days, some left, and 12 men remained. The number of men who left is the difference between the initial number of men and the number of men who remained.

Number of Men Who Left = Initial Number of Men - Number of Remaining Men

Number of Men Who Left = \(30 \text{ men} - 12 \text{ men} = 18 \text{ men}\)

Therefore, 18 men left the job after 24 days.

Conclusion

Based on the calculations, the number of men who left the job is 18.

Revision Table: Work and Time Concepts

Understanding the relationship between work, men, and time is crucial for these types of problems.

• Work Done: Work is often measured in 'man-days' or 'man-hours', representing the total effort needed.
• Formula: Work = Number of Workers × Time Taken
• Inverse Proportion: If the amount of work is constant, the number of workers is inversely proportional to the time taken. More workers mean less time, and fewer workers mean more time.
• Partial Work: If work is done in stages or by changing numbers of workers, calculate the work done in each stage and the remaining work.

Additional Information: Types of Work and Time Problems

Work and Time problems in quantitative aptitude can come in several forms:

• Problems involving a fixed number of workers and the time taken.
• Problems involving changes in the number of workers (men joining or leaving).
• Problems involving workers with different efficiencies.
• Problems involving alternating work days between workers.
• Problems related to pipes and cisterns (which are analogous to work and time).

Solving these problems often involves calculating the total work unit (like man-days), the work done in different phases, and then using the remaining work to find the unknown variable (like time or number of workers).