Understanding the Work and Time Problem
This problem asks for the minimum time needed to complete a work by four individuals, A, B, C, and D, each having a different working speed. We must also consider the specific rules about how they work: only one person works per hour, and no one can work in two consecutive hours.
Calculating Individual Work Rates
First, we determine the rate at which each person works. The rate is the fraction of the work completed per hour. It's calculated as \(1 / (\text{time taken to complete the work})\).
- Person A's rate: A completes the work in 3 hours, so A's rate is \( \frac{1}{3} \) work per hour.
- Person B's rate: B completes the work in 6 hours, so B's rate is \( \frac{1}{6} \) work per hour.
- Person C's rate: C completes the work in 9 hours, so C's rate is \( \frac{1}{9} \) work per hour.
- Person D's rate: D completes the work in 12 hours, so D's rate is \( \frac{1}{12} \) work per hour.
Analyzing the Working Constraints
The constraints are crucial for finding the minimum time:
- Single Worker Per Hour: Only one person can perform work during any single hour.
- No Consecutive Work: An individual cannot work in one hour and then immediately work in the next hour. They must have at least one hour break between their working shifts.
- Optional Engagement: It's not required for all four people (A, B, C, D) to work; we can use only those needed to finish fastest.
Our goal is to complete 1 unit of work in the shortest possible total time.
Strategy for Minimum Time Calculation
To minimize the total time, we should prioritize using the workers who are the most efficient (have the highest work rates). Person A is the fastest (\(1/3\) work/hour). However, due to the constraint that no one can work for two consecutive hours, A cannot work continuously.
The best strategy is to pair the most efficient worker (A) with the next most efficient worker available, ensuring the constraint is met. Let's consider pairing A with B, the second fastest worker.
We can analyze the work done in a 2-hour cycle:
- Hour 1: Person A works. Work done = \( \frac{1}{3} \).
- Hour 2: Person A cannot work again. The next best choice is Person B. Work done = \( \frac{1}{6} \).
The total work completed in these 2 hours (1 hour by A, 1 hour by B) is:
\( \text{Work done in 2 hours} = \text{A's rate} + \text{B's rate} = \frac{1}{3} + \frac{1}{6} \)To add these fractions, we use a common denominator, which is 6:
\( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \)This means that by alternating A and B, they complete \( \frac{1}{2} \) of the work every 2 hours.
Determining Total Minimum Hours
To complete the entire work (1 unit), we need to find out how many of these 2-hour cycles are needed.
Total work = 1 unit.
Work completed per 2-hour cycle = \( \frac{1}{2} \) unit.
Number of cycles required = \( \frac{\text{Total Work}}{\text{Work per cycle}} = \frac{1}{1/2} = 2 \) cycles.
Since each cycle takes 2 hours, the total minimum time is:
\( \text{Total Minimum Time} = \text{Number of cycles} \times \text{Hours per cycle} \) \( \text{Total Minimum Time} = 2 \times 2 \text{ hours} = 4 \text{ hours} \)The sequence of work would be: A works (Hour 1), B works (Hour 2), A works (Hour 3), B works (Hour 4). This completes the work in exactly 4 hours and adheres to all the rules.
We can confirm this is the minimum by noting that any other pairing (like A with C, or B with C) would result in less work done per 2-hour cycle, leading to a longer total time.
For example:
- A + C in 2 hours = \( \frac{1}{3} + \frac{1}{9} = \frac{4}{9} \) work. Total time = \( \frac{1}{4/9} \times 2 = \frac{9}{2} = 4.5 \) hours.
- A + D in 2 hours = \( \frac{1}{3} + \frac{1}{12} = \frac{5}{12} \) work. Total time = \( \frac{1}{5/12} \times 2 = \frac{12}{5} \times 2 = 4.8 \) hours.
Therefore, the minimum number of hours required is 4.
Similar Questions
- 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?
- 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?
