Understanding the Work Done Relationship
This problem involves the concept of work done, which is typically calculated as the product of the number of workers, the time spent working, and the rate of work. Assuming the rate of work for each man is the same, the total work done is directly proportional to the number of men and the number of days they work.
We can express this relationship as:
Work = (Number of Men) × (Number of Days)
Men and Days Equation Setup
Let's denote the number of men and days for the two scenarios:
- Scenario 1: \(x\) men working for \((x + 1)\) days.
- Scenario 2: \((x + 5)\) men working for \((x - 2)\) days.
The problem states that the work done in both scenarios is equal. Therefore, we can set up the equation:
Work1 = Work2
\( x \times (x + 1) = (x + 5) \times (x - 2) \)
Solving the Work Equation Algebraically
Now, we need to solve this equation for \(x\).
- Expand both sides of the equation:
- Left side: \( x \times (x + 1) = x^2 + x \)
- Right side: \( (x + 5) \times (x - 2) = x(x - 2) + 5(x - 2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10 \)
- Set the expanded expressions equal:
\( x^2 + x = x^2 + 3x - 10 \)
- Simplify the equation: Subtract \( x^2 \) from both sides:
\( x = 3x - 10 \)
- Isolate \(x\): Rearrange the terms to solve for \(x\). Subtract \(x\) from both sides and add 10 to both sides:
\( 10 = 3x - x \)
\( 10 = 2x \)
- Calculate the value of \(x\): Divide by 2:
\( x = \frac{10}{2} \)
\( x = 5 \)
Validating the Solution
We must ensure that the values for men and days are physically possible (i.e., positive numbers).
If \( x = 5 \):
- Number of men in scenario 1 = \( x = 5 \) (Positive)
- Number of days in scenario 1 = \( x + 1 = 5 + 1 = 6 \) (Positive)
- Number of men in scenario 2 = \( x + 5 = 5 + 5 = 10 \) (Positive)
- Number of days in scenario 2 = \( x - 2 = 5 - 2 = 3 \) (Positive)
All values are positive, so \( x = 5 \) is a valid solution.
Let's check the work done:
- Work1 = \( 5 \text{ men} \times 6 \text{ days} = 30 \) units of work.
- Work2 = \( 10 \text{ men} \times 3 \text{ days} = 30 \) units of work.
The work done is indeed equal in both scenarios.
Conclusion
Based on the algebraic solution and validation, the value of \(x\) is 5.
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?
- 50 men can complete a work in 40 days. They begin the work together but a batch of 5 men left after each period of 10 days. What is the time to complete the work?
- \(p\) 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?
