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 together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
Two men and 7 women can complete a work in 28 days whereas 6 men and 16 women can do the same work in 11 days. In how many days can 7 men complete the same work?
