Question

25 men can complete a job in 30 days. After how many days should the strength of work force be increased by 50 men 50 that the work will be completed in \(\frac{2}{3}\) rd of the actual time: (a) 15 (b) 10 (c) 18 (d) 5

Short answer

Answer: 10 days

Step-by-step solution

Find the work completed by 25 men in 1 day

We know that 25 men complete the job in 30 days. Therefore, the work done by them in one day can be calculated as: Work by 25 men in 1 day = \(\frac{1 \text{ job}}{30}\) So we get the equation: 25 men × 1 day = \(\frac{1 \text{ job}}{30}\)

Total work needed to complete the job in \(\frac{2}{3}\)rd of the actual time

We are given that the work should be completed in \(\frac{2}{3}\) rd of the actual time. The actual time to complete the job is 30 days, so we have to find the work that should be completed in \(\frac{2}{3}\) × 30 days, which is 20 days: Total work needed to be completed = Work by 25 men × 20 days As work by 25 men in 1 day is \(\frac{1 \text{ job}}{30}\), we get: Total work needed = 25 men × 20 days × \(\frac{1 \text{ job}}{30}\)

Calculate the work done by 25 men and 75 men in 'x' days

Let's assume it takes 'x' days before the strength of the workforce needs to be increased by 50 men. So, in these 'x' days, work will be done by 25 men. After 'x' days, there will be a total of 75 men (25 original men + 50 additional men), and they will work for the remaining (20-x) days. Work done by 25 men in 'x' days = 25 men × 'x' days × \(\frac{1 \text{ job}}{30}\) Work done by 75 men in (20-x) days = 75 men × (20-x) days × \(\frac{1 \text{ job}}{30}\)

Set up the equation and solve for 'x'

Since the work done by 25 men in 'x' days and the work done by 75 men in (20-x) days must be equal to the total work needed to finish the job in \(\frac{2}{3}\) rd of the time, we can set up the equation: 25 x \(\frac{1}{30}\) x + 75 x \(\frac{1}{30}\) (20-x) = 25 x \(\frac{1}{30}\) x 20 Now we can solve for x: x + 3(20-x) = 20 Simplify and solve: x + 60 - 3x = 20 -2x = -40 x = 20 So, the strength of the workforce should be increased by 50 men after 10 days. Therefore, the correct answer is (b) 10.

Key concepts

time management

When tackling any job, particularly extensive ones, time management is crucial. It helps in determining the pace of work and how resources should be allocated throughout the project's duration.
In this context, understanding how long a job should take if it needs to be finished by a certain time is vital.

Time management starts by visualizing the complete timeline of the task. Here, the job needed to be completed in \(\frac{2}{3}\) of the original planned time. From a time management perspective:

• Identifying the full duration required (in this case, initially 30 days).
• Determining the new target timeframe (20 days, being \(\frac{2}{3}\) of the initial period).

Effective time management involves setting these time constraints early on and using them to plan how resources will be managed and distributed effectively to finish within the new timeline.

work calculation

Work calculation refers to understanding how much of the job is completed over a specific period.
For instance, it requires calculating the amount of work done by 25 men in a day. We can extract vital information about work progress using simple formulas.

In our scenario, we calculate:

• The daily contribution or rate of work completion (\(\frac{1 \text{ job}}{30}\) per day by 25 men).
• The total work required over any period set by time management (here, 20 days to meet the revised deadline).

These calculations enable project managers to precisely align the workforce and set realistic expectations for project completion.

mathematical problem-solving

Mathematical problem-solving is at the heart of determining the solution to this problem.
It involves setting up equations and solving them logically.

These are the problem-solving steps:

• Set the goal: Complete the work in 20 days.
• Determine initial workforce contribution: 25 men for 'x' days.
• Adjust for a faster completion per new plan: Increase workforce by 50 men after 'x' days.

By representing these real-world conditions with equations, one can solve for 'x' (the number of days until the workforce increase needs to happen). Here, solving the balance equation fulfills the need to match the accelerated work rate with the changed timeline.

efficiency improvement

Efficiency improvement in workforce optimization can be achieved by modifying how the available resources are used.
This involves strategies that boost productivity without overly extending the timeline or straining the resources.

In the exercise, improving efficiency meant increasing manpower to maintain the job's pace:

• Initial efficiency: 25 men work at a pace over a static 30 days.
• Desired efficiency: Add 50 additional workers after evaluating progress (10 days), aiming to finish in 20 days.

Efficiency improvements ensure that when conditions or desired outcomes change (like completing sooner), the workforce adapts by both varying the number of workers and the timing of their input to meet new demands. This kind of flexible planning and response is key to maintaining high productivity and meeting project goals.