Question

30 workers can finish a work in 20 days. After how many dajs should 9 workers leave the job so that the work is completed in total 26 days : (a) 12 (b) 10 (c) 6 (d) none of these

Short answer

Answer: 6 days

Step-by-step solution

Calculate the total work

The total work done can be represented as the product of the number of workers and the number of days taken to complete the work. In this case, 30 workers can finish the work in 20 days. So, the total work can be represented as: Total work = 30 workers * 20 days

Calculate the work done before 9 workers leave

Let's assume that after 'x' days, 9 workers leave the job. So, for the first 'x' days, all 30 workers are working. The work done during this period can be represented as: Work done by 30 workers in 'x' days = 30 workers * x days

Calculate the work done by the remaining workers

After 'x' days, 9 workers leave the job, leaving 21 workers (30-9 = 21) to complete the work. They take the remaining days (26 days - x days) to complete the work. The work done by the remaining 21 workers can be represented as: Work done by 21 workers in (26-x) days = 21 workers * (26 - x) days

Calculate the equation for the total work

The sum of the work done by the 30 workers in the first 'x' days and the work done by the remaining 21 workers in the (26 - x) days should equal the total work. Therefore, we can set up an equation as follows: (30 workers * x days) + (21 workers * (26 - x) days) = 30 workers * 20 days

Solve the equation for 'x'

Now, we'll solve the equation to get the value of 'x': 30x + 21(26 - x) = 30 * 20 30x + 546 - 21x = 600 9x = 54 x = 6

Conclusion

The value of 'x' is 6, which means that after 6 days, 9 workers should leave the job so that the work is completed in a total of 26 days. So, the correct answer is (c) 6.

Key concepts

Number of Workers

Understanding the role of the number of workers in completing a task is essential in work-related problems. In the given exercise, we begin with 30 workers. The number of workers directly influences how quickly work can be completed. The more workers there are, the faster they can finish a task. Conversely, with fewer workers, the task takes longer.
To grasp this better, think of each worker contributing to a portion of the work daily. If each worker can complete one unit of work per day, 30 workers will finish 30 units in a day. When the number of workers decreases, such as when 9 workers leave, only the remaining 21 workers are left to perform the task. Be attentive to changes in worker numbers, as it directly impacts the length of time needed to achieve the same total work.

Total Work Calculation

When dealing with work and time problems, calculating the total work is crucial. In our problem, total work is the product of the initial number of workers and the total days they would have taken to complete the task without interruptions.
In this exercise, 30 workers can finish the work in 20 days. Therefore, the total work is calculated by:\[\text{Total work} = 30 \times 20 = 600 \text{ worker-days}\]This equation helps us understand how much work (in terms of worker-days) the job represents. Worker-days is a way to measure the work considering both the number of workers and the time they spend on the job. This value remains constant unless more work is added or removed.

Days Calculation

Calculating the number of days required to finish a task involves understanding the interplay between workers present and the total work required. From the problem, we need to determine after how many days 9 workers should leave to finish the task in 26 days.
Initially, we have 30 workers. Let's say they work fully for \(x\) days. They accomplish a portion of the work, calculated by:\[\text{Work by 30 workers in } x \text{ days} = 30x \]After \(x\) days, 9 workers leave, leaving 21 workers. These 21 workers continue for \(26-x\) days:\[\text{Work by 21 workers in } (26-x) \text{ days} = 21(26-x)\]All parts of the job done by all workers should combine to achieve the total work:\[30x + 21(26 - x) = 600\]Solving this equation, we find:\[30x + 546 - 21x = 600 \ 9x = 54 \ x = 6\]Therefore, 9 workers should leave after 6 days to ensure the task completes in 26 days.