Work Problem: Calculating Time with Decreasing Workforce
This problem involves calculating the total time needed to complete a work when the number of workers changes over time. We are given the initial number of men, the time they would take if they all worked together throughout, and the condition that some men leave periodically.
Understanding Total Work Required
First, let's determine the total amount of work that needs to be done. This is often measured in 'man-days', which represents the total effort required.
Total Work = (Initial Number of Men) × (Initial Time)
Given:
- Initial Number of Men = 50
- Initial Time = 40 days
Using LaTeX for the formula:
Total Work = \(50 \text{ men} \times 40 \text{ days} = 2000 \text{ man-days}\)
So, the project requires a total of 2000 man-days of effort.
Step-by-Step Work Calculation with Workforce Reduction
The problem states that 5 men leave after *each period* of 10 days. We need to calculate the work done in each 10-day interval and track the remaining workforce and work.
Period 1: Days 1-10
- Number of men working = 50
- Work done in this period = \(50 \text{ men} \times 10 \text{ days} = 500 \text{ man-days}\)
- Total work done so far = 500 man-days
- Remaining work = \(2000 - 500 = 1500 \text{ man-days}\)
- At the end of day 10, 5 men leave.
- Number of men for the next period = \(50 - 5 = 45\)
Period 2: Days 11-20
- Number of men working = 45
- Work done in this period = \(45 \text{ men} \times 10 \text{ days} = 450 \text{ man-days}\)
- Total work done so far = \(500 + 450 = 950 \text{ man-days}\)
- Remaining work = \(2000 - 950 = 1050 \text{ man-days}\)
- At the end of day 20, 5 men leave.
- Number of men for the next period = \(45 - 5 = 40\)
Period 3: Days 21-30
- Number of men working = 40
- Work done in this period = \(40 \text{ men} \times 10 \text{ days} = 400 \text{ man-days}\)
- Total work done so far = \(950 + 400 = 1350 \text{ man-days}\)
- Remaining work = \(2000 - 1350 = 650 \text{ man-days}\)
- At the end of day 30, 5 men leave.
- Number of men for the next period = \(40 - 5 = 35\)
Period 4: Days 31-40
- Number of men working = 35
- Work done in this period = \(35 \text{ men} \times 10 \text{ days} = 350 \text{ man-days}\)
- Total work done so far = \(1350 + 350 = 1700 \text{ man-days}\)
- Remaining work = \(2000 - 1700 = 300 \text{ man-days}\)
- At the end of day 40, 5 men leave.
- Number of men for the next period = \(35 - 5 = 30\)
Period 5: Days 41 onwards
- Number of men working = 30
- Remaining work to be done = 300 man-days
- Time needed to complete the remaining work = \(\frac{\text{Remaining Work}}{\text{Number of men}}\)
- Time needed = \(\frac{300 \text{ man-days}}{30 \text{ men}} = 10 \text{ days}\)
Total Time Calculation
The total time to complete the work is the sum of the time taken in all periods until the work is finished.
Total Time = (Time for Period 1) + (Time for Period 2) + (Time for Period 3) + (Time for Period 4) + (Time needed for remaining work)
Total Time = 10 days + 10 days + 10 days + 10 days + 10 days
Total Time = 50 days
Based on the standard calculation method for work problems, the total time required to complete the work is 50 days. The calculation involves determining the total work required and then accounting for the decreasing number of workers over successive 10-day periods.
Similar Questions
- A, B, C, D can complete a work in 3, 6, 9, 12 hours respectively. Further, only one person can work at a time in each hour and nobody can work for two consecutive hours. It is not necessary to engage all. What is the minimum number of hours that they will take to finish the work?
- If the work done by \(x\) men in \((x + 1)\) days is equal to the work done by \((x + 5)\) men in \((x - 2)\) days, then what is the value of \(x\)?
- \(p\) number of men can finish a piece of work in \(q\) days. If there are 50% more men, then the work will be finished 12 days earlier. What is the value of \(q\)?
- Eight men and 24 women can finish a piece of work in one day while 12 men and 18 women can also finish it in one day. What is the time taken by 24 men and 72 women to finish the work ?
- X can do a piece of work in 4 days and Y can do the same piece of work in 6 days. Which of the following statements is/are correct ?
I. If X and Y work alternately starting with X, then the piece of work will be finished on the fifth day.
II. If X and Y work alternately starting with Y, then the piece of work will be finished in less than 5 days.
Select the answer using the code given below : - 12 men working 8 hours a day can completely build a wall of length 100 m, breadth 20 cm and height 5 m in 10 days. How many days will 16 men working 10 hours a day require to build a wall of length 200 m, breadth 60 cm and height 5 m ?
Important Questions from Time and work
A person can complete 20% of work in 8 days and another person y can complete 25% of the same work in 6 days. If they work together, in how many days will 40% of the work be completed?
P works thrice as fast as Q, whereas P and Q together can work four times as fast as R. If P, Q and R together work on a job, in what ratio should they share the earnings?
There are three pillars X, Y and Z of different heights. Three spiders A, B and C start to climb on these pillars simultaneously. In one chance, A climbs on X by 6 cm but slips down 1 cm. B climbs on Y by 7 cm but slips down 3 cm. C climbs on Z by 6.5 cm but slips down 2 cm. If each of them requires 40 chances to reach the top of the pillars, what is the height of the shortest pillar?
There is an order of 19000 quantity of a particular product from a customer. The firm produces 1000 quantity of that product per out of which 5% are unfit for sale. In how many days will the order be completed?
Ram and Shyam work on a job together for four days and complete 60% of it. Ram takes leave then and Shyam works for eight more days to complete the job. How long would Ram take to complete the entire job alone?
