A work when done by 10 women is completed in 12 days. The same work can be completed in 8 days by 5 men. How many days will it take to complete when 6 women and 3 men are employed to perform the same job?
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Understanding Work and Time Problems
Work and Time problems often involve calculating how long it takes individuals or groups to complete a task based on their work rates. The fundamental principle is that if someone completes a work in 'n' days, their work rate per day is \(\frac{1}{n}\) of the total work. When multiple people work together, their individual work rates are added to find the combined work rate.
Calculating Individual Work Rates
We are given the following information:
- 10 women can complete the work in 12 days.
- 5 men can complete the same work in 8 days.
Let's find the work rate of one woman and one man per day.
- Work rate of 10 women: If 10 women complete the work in 12 days, the total work done by 10 women in one day is \(\frac{1}{12}\) of the total work.
- Work rate of 1 woman: To find the work rate of a single woman, we divide the work rate of 10 women by 10.
Work rate of 1 woman per day \(= \frac{1}{12} \div 10 = \frac{1}{12} \times \frac{1}{10} = \frac{1}{120}\) of the total work. - Work rate of 5 men: If 5 men complete the work in 8 days, the total work done by 5 men in one day is \(\frac{1}{8}\) of the total work.
- Work rate of 1 man: To find the work rate of a single man, we divide the work rate of 5 men by 5.
Work rate of 1 man per day \(= \frac{1}{8} \div 5 = \frac{1}{8} \times \frac{1}{5} = \frac{1}{40}\) of the total work.
So, a woman completes \(\frac{1}{120}\) of the work per day, and a man completes \(\frac{1}{40}\) of the work per day.
Calculating Combined Work Rate
Now, we need to find out how many days it will take for 6 women and 3 men to complete the job when working together. First, we calculate their combined work rate per day.
- Work done by 6 women in one day: This is 6 times the work rate of one woman.
Work by 6 women per day \(= 6 \times \frac{1}{120} = \frac{6}{120} = \frac{1}{20}\) of the total work. - Work done by 3 men in one day: This is 3 times the work rate of one man.
Work by 3 men per day \(= 3 \times \frac{1}{40} = \frac{3}{40}\) of the total work. - Combined work rate of 6 women and 3 men: Add the work done by 6 women and 3 men in one day.
Combined work rate per day \(= (\text{Work by 6 women}) + (\text{Work by 3 men})\)
Combined work rate per day \(= \frac{1}{20} + \frac{3}{40}\)
To add these fractions, we find a common denominator, which is 40.
\(\frac{1}{20} = \frac{1 \times 2}{20 \times 2} = \frac{2}{40}\)
So, the combined work rate per day \(= \frac{2}{40} + \frac{3}{40} = \frac{2 + 3}{40} = \frac{5}{40}\)
This fraction can be simplified:
Combined work rate per day \(= \frac{5}{40} = \frac{1}{8}\) of the total work.
Determining Time Taken
If 6 women and 3 men together complete \(\frac{1}{8}\) of the total work in one day, the total number of days required to complete the entire work (which is 1 unit of work) is the reciprocal of their combined work rate.
Time taken \(= \frac{\text{Total Work}}{\text{Combined Work Rate per Day}}\)
Time taken \(= \frac{1}{\frac{1}{8}} = 1 \times 8 = 8\) days.
Therefore, it will take 8 days for 6 women and 3 men to complete the same job.
Summary of Calculations
| Group | Time Taken (days) | Total Work per Day | Work Rate per Person per Day |
|---|---|---|---|
| 10 Women | 12 | \(\frac{1}{12}\) | \(\frac{1}{120}\) (per woman) |
| 5 Men | 8 | \(\frac{1}{8}\) | \(\frac{1}{40}\) (per man) |
| 6 Women | - | \(6 \times \frac{1}{120} = \frac{1}{20}\) | - |
| 3 Men | - | \(3 \times \frac{1}{40} = \frac{3}{40}\) | - |
| 6 Women + 3 Men | ? | \(\frac{1}{20} + \frac{3}{40} = \frac{1}{8}\) | - |
Since the combined work rate is \(\frac{1}{8}\) of the work per day, they will complete the work in 8 days.
Revision Table: Key Concepts in Work and Time
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | The amount of work done by a person or group in a unit of time (e.g., per day, per hour). | Work Rate \(= \frac{\text{Total Work}}{\text{Time Taken}}\) |
| Time Taken | The total time required to complete a certain amount of work. | Time Taken \(= \frac{\text{Total Work}}{\text{Work Rate}}\) |
| Combined Work Rate | The sum of individual work rates when multiple people or groups work together. | Combined Work Rate \(= \text{Rate}_1 + \text{Rate}_2 + \text{Rate}_3 + ...\) |
| Total Work | Often considered as 1 unit of work for calculation purposes. | Total Work \(= \text{Work Rate} \times \text{Time Taken}\) |
Additional Information: Efficiency and Inverse Relationship
Work and time problems often involve the concept of efficiency. Efficiency is directly proportional to the work rate. A more efficient person does more work in the same amount of time or completes the same work in less time.
There is an inverse relationship between the number of workers and the time taken to complete a job, assuming all workers have the same efficiency. If you increase the number of workers, the time taken to complete the same amount of work decreases. Similarly, time taken is inversely proportional to the work rate.
In this problem, we first found the individual work rates based on the given group rates and times. Then, we used these individual rates to find the combined rate of the new group (6 women and 3 men) and finally calculated the time required.
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