15 men can complete a work in 25 days, and 25 women can complete the same work in 40 days. If all 15 men and 25 women work together, in how many days will the work get completed?
Solving Work and Time Problems
This problem involves calculating the combined work rate of men and women to find out how long it takes them to complete a task together. We are given the time it takes for a certain number of men and a certain number of women to complete the same work individually. We need to find the time taken when both groups work simultaneously.
Calculating Individual Work Rates
First, let's determine the amount of work a single person or a group can do in one day. This is often called their work rate.
- We know 15 men can complete the work in 25 days.
- This means that in 1 day, 15 men complete \(\frac{1}{25}\) of the total work.
Similarly, for the women:
- We know 25 women can complete the same work in 40 days.
- This means that in 1 day, 25 women complete \(\frac{1}{40}\) of the total work.
Calculating Combined Work Rate
When 15 men and 25 women work together, their individual work rates for one day add up to give the combined work rate for one day.
Combined work done by 15 men and 25 women in 1 day = (Work done by 15 men in 1 day) + (Work done by 25 women in 1 day)
Combined work rate = \( \frac{1}{25} + \frac{1}{40} \)
To add these fractions, we need to find a common denominator. The least common multiple (LCM) of 25 and 40 is 200.
Convert the fractions to have the denominator 200:
- \( \frac{1}{25} = \frac{1 \times 8}{25 \times 8} = \frac{8}{200} \)
- \( \frac{1}{40} = \frac{1 \times 5}{40 \times 5} = \frac{5}{200} \)
Now, add the fractions:
\( \text{Combined work rate} = \frac{8}{200} + \frac{5}{200} = \frac{8 + 5}{200} = \frac{13}{200} \)
So, together, 15 men and 25 women complete \(\frac{13}{200}\) of the work in 1 day.
Calculating Total Time Taken
If \(\frac{13}{200}\) of the work is done in 1 day, then the total number of days required to complete the entire work (which is 1 whole unit of work) is the reciprocal of the combined work rate.
Time taken = \( \frac{1}{\text{Combined work rate}} = \frac{1}{\frac{13}{200}} = \frac{200}{13} \) days.
To express this as a mixed number, we divide 200 by 13:
\( 200 \div 13 \)
\( 200 = 13 \times 15 + 5 \)
So, the time taken is \( 15 \frac{5}{13} \) days.
Therefore, if all 15 men and 25 women work together, the work will be completed in \( 15 \frac{5}{13} \) days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | The amount of work done by a person or group in one unit of time (e.g., per day). | Work Rate = \( \frac{\text{Total Work}}{\text{Time Taken}} \) |
| Total Work | Often considered as 1 unit for a single task. | Total Work = Work Rate \(\times\) Time Taken |
| Time Taken | The total duration required to complete the work. | Time Taken = \( \frac{\text{Total Work}}{\text{Work Rate}} \) |
| Combined Work Rate | Sum of individual work rates when multiple people/groups work together. | Combined Rate = Rate\(_{1}\) + Rate\(_{2}\) + ... |
Additional Information: Work and Time Variations
Work and time problems often involve variations like:
- Individual work: Finding the time taken by a single person when a group's time is known.
- Efficiency: Comparing the work rates of different individuals or groups. If person A is twice as efficient as person B, A's work rate is double B's.
- Men, Women, Children problems: Relating the work rate of men, women, and children (e.g., 2 men = 3 women in terms of work).
- Work completed in parts: Problems where different groups work for specific periods or complete portions of the work.
- Pipes and Cisterns: Similar concept where pipes filling a tank are like positive work rates, and pipes emptying are like negative work rates.
Understanding the concept of work rate (work done per unit time) is key to solving these types of problems. The total work is usually treated as '1 unit'.
Similar Questions
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?
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:
15 men and 25 women can complete a piece of work in 9.6 days. If 16 women can complete the same work in 27 days, find the number of days in which 16 men can complete the same work.
R, S and T can finish a work in 20, 15 and 10 days, respectively. R works on all days and S and T work on alternate days with T starting the work on the first day. In how many days is the work finished?
A can do \(1 \over 3\) of a piece of work in 32 days, B can do \(37{1 \over 2}\)% of the same work in 24 days, while C can do 60% of the same work in 48 days. B and C together started and worked for x days. After x days, B left the work and A joined C and both completed the remaining work in (x + 8) days. If the ratio of the work done by (B + C) together to the work done by (A + C) together is 9 ∶ 11, then what fraction of the same work can be completed by C alone in 3.5x days?
A group of college students had decided to complete a project in 10 days. As 2 students dropped out every day, the project got completed at the end of the 15th day. The number of students at the beginning of the project was:
Aarif, Arun and Abraham can do a work in 12, 20 and 24 days, respectively. They all begin together. Arun leaves the work 3 days and Abraham 6 days before its completion. In how many days is the work finished?
A,B and C can do a piece of work in 30 days, 40 days and 50 days, respectively. Beginning with A, if A, B and C do the work alternatively then in how many days will the work be finished?
Working 5 hours a day, A can complete a task in 8 days and working 6 hours a day, B can finish the same task in 10 days, working 8 hours a day, they can jointly complete the task in __________.
A can do 20% of a job in 7 days and B can do 25% of the job in 7 days if they worked alone. How much of the job (in percentage) can they complete in 7 days if they worked together?
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?
