Work and Time Calculation
This problem involves calculating the time required for a combined group of men and women to complete a work, given the time taken by individual groups.
Equating Work Capacity
We are given that 10 men can complete the work in 170 days, and 15 women can complete the same work in 170 days. This implies that the work done by 10 men is equivalent to the work done by 15 women.
Let 'M' represent the work rate of one man and 'W' represent the work rate of one woman.
From the problem statement:
- \( 10 \times M \times 170 = \text{Total Work} \)
- \( 15 \times W \times 170 = \text{Total Work} \)
Equating the work done:
\( 10 M \times 170 = 15 W \times 170 \)
This simplifies to:
\( 10 M = 15 W \)
Dividing both sides by 5:
\( 2 M = 3 W \)
This relationship tells us the relative work efficiency of men and women. From this, we can express the work rate of a woman in terms of a man's work rate:
\( W = \frac{2}{3} M \)
Calculating Combined Workforce Efficiency
We need to find the time taken by 20 men and 20 women working together.
First, let's express the combined workforce in terms of a single unit, say, equivalent men.
The combined group consists of 20 men and 20 women. Substitute the equivalent work rate of women:
\( \text{Total Equivalent Men} = 20 M + 20 W \)
\( = 20 M + 20 \left( \frac{2}{3} M \right) \)
\( = 20 M + \frac{40}{3} M \)
\( = \left( \frac{60}{3} + \frac{40}{3} \right) M \)
\( = \frac{100}{3} M \)
So, 20 men and 20 women together have the work capacity equivalent to \( \frac{100}{3} \) men.
Calculating Total Time Required
The total amount of work can be calculated from the initial condition. Let's use the men's data:
\( \text{Total Work} = 10 \text{ men} \times 170 \text{ days} = 1700 \text{ man-days} \)
Now, we can find the time required for the combined group (\( \frac{100}{3} \) men) to complete this work:
\( \text{Time} = \frac{\text{Total Work}}{\text{Number of Equivalent Men}} \)
\( \text{Time} = \frac{1700 \text{ man-days}}{\frac{100}{3} \text{ men}} \)
\( \text{Time} = 1700 \times \frac{3}{100} \text{ days} \)
\( \text{Time} = 17 \times 3 \text{ days} \)
\( \text{Time} = 51 \text{ days} \)
Alternatively, we could convert the combined workforce to equivalent women:
\( 2 M = 3 W \implies M = \frac{3}{2} W \)
\( \text{Total Equivalent Women} = 20 M + 20 W = 20 \left( \frac{3}{2} W \right) + 20 W \)
\( = 30 W + 20 W = 50 W \)
\( \text{Total Work} = 15 \text{ women} \times 170 \text{ days} = 2550 \text{ woman-days} \)
\( \text{Time} = \frac{2550 \text{ woman-days}}{50 \text{ women}} = \frac{255}{5} \text{ days} = 51 \text{ days} \)
Conclusion
Both methods confirm that 20 men and 20 women will take 51 days to complete the same work.
Similar Questions
- A contractor employs 15 men to complete a work in 24 days, working 8 hours per day. After six days, he adds five more men but also reduces the working hours to six per day from the seventh day onwards. If all men work with the same efficiency throughout, how many more days will be required to finish the remaining work?
- 6 women and 4 men can complete a work in 10 days, while 4 women and 5 men complete it in 10 days. In how many days will 8 women complete it?
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Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
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 will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
