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
A and B working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
