A can do a work in 6 days, which B alone can do in 8 days. In how many days they together can do it
Solving Work and Time Problems: A and B Together
This question asks us to find out how many days it will take for A and B to complete a certain work if they work together. We are given the time each person takes individually to finish the same work.
Understanding Work Rate
In problems involving work and time, it's helpful to think about the 'work rate' of each person. Work rate is the amount of work done by a person in one day (or one unit of time).
- If a person can do a work in \(d\) days, then their work rate is \(\frac{1}{d}\) of the work per day.
Calculating Individual Work Rates
- A can do the work in 6 days.
- A's work rate = \(\frac{1}{6}\) of the work per day.
- B can do the work in 8 days.
- B's work rate = \(\frac{1}{8}\) of the work per day.
Calculating Combined Work Rate
When A and B work together, their work rates add up. To find their combined work rate, we add their individual work rates:
Combined work rate = A's work rate + B's work rate
Combined work rate = \(\frac{1}{6} + \frac{1}{8}\)
To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 8 is 24.
Combined work rate = \(\frac{1 \times 4}{6 \times 4} + \frac{1 \times 3}{8 \times 3}\)
Combined work rate = \(\frac{4}{24} + \frac{3}{24}\)
Combined work rate = \(\frac{4 + 3}{24}\)
Combined work rate = \(\frac{7}{24}\) of the work per day.
Calculating Time Taken Together
The total time taken to complete the work when working together is the reciprocal of the combined work rate.
Time taken together = \(\frac{1}{\text{Combined work rate}}\)
Time taken together = \(\frac{1}{\frac{7}{24}}\)
Time taken together = \(\frac{24}{7}\) days.
So, A and B together can complete the work in \(\frac{24}{7}\) days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula/Relationship |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | Work Rate = \( \frac{1}{\text{Time Taken}} \) |
| Total Work | Usually considered as 1 unit or the LCM of individual times. | Total Work = Rate \(\times\) Time |
| Combined Rate | Sum of individual work rates when people work together. | Rate\(_\text{A+B}\) = Rate\(_\text{A}\) + Rate\(_\text{B}\) |
| Time Together | Time taken by individuals working together to complete the work. | Time\(_\text{A+B}\) = \( \frac{1}{\text{Rate}_\text{A+B}} \) |
Additional Information on Work and Time Problems
Work and time problems often involve finding how long it takes individuals or groups to complete a task. The key idea is to convert the given time into a 'rate' at which the work is done per unit of time. When people work together, their individual rates of working are usually added up to find the combined rate.
Sometimes problems might involve:
- More than two people working together.
- People working for different amounts of time.
- Some people leaving or new people joining the work.
- Negative work (like a leak filling a tank while another pipe empties it).
In all these cases, the fundamental principle of using work rates remains the same. Calculate the rate for each part of the problem and then use the relationship: Work = Rate \(\times\) Time.
Important Questions from Work Efficiency
- M is thrice as efficient as workman as N and therefore is able to finish a job in 264 days less than N. Then in how many days can N finish the entire work?
- A can do a piece of work in $1455$ days. B is $25\%$ more efficient than A. Find the number of days taken by B to do the same piece of work.
- 45 people take 18 days to dig a pond. If the pond would have to be dug in 15 days, then the number of people to be employed will be
- A stock of food is enough for 240 men for 48 days. How long will the same stock last for 160 men?
One contractor had undertaken a construction work. To complete the given job, he sought labour comprising 24 men for 12 days for working 10 hours a day. But due to festival season, as the labour had gone to their respective hometowns, he could get only 15 men and they too were willing to work 8 hours a day. With this much labour, in how many days would he be able to complete the same job?
