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
Five men and 2 boys can do in 30 days as much work as 7 men and 10 boys can do in 15 days. How many boys should join 40 men to do the same work in 4 days?
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work further increased by 50%?
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
A and B together can do a piece of work in 4 days, B and C can do it in 6 days, A and C can do it in 8 days. Then A, B and C together can do the same work in :-
A can complete 50% of a work in 9 days and B can do 25% of the work in 9 days, if they work alone. If they work together then how much work (in percentage) can be completed in 6 days?
