Work Rate Calculation
The problem asks for the time 3 women and 6 men need to complete a work, given the time taken by 4 men (18 days) and 9 women (8 days).
Individual Work Rates Analysis
Let $W$ represent the total amount of work to be completed.
Men's Rate:
4 men complete the work $W$ in 18 days.
The work done per day by 4 men is $\frac{W}{18}$.
Therefore, the work done per day by 1 man is $\frac{W}{4 \times 18} = \frac{W}{72}$.
The work rate of 1 man is $\frac{1}{72}$ of the work per day.
Women's Rate:
9 women complete the work $W$ in 8 days.
The work done per day by 9 women is $\frac{W}{8}$.
Therefore, the work done per day by 1 woman is $\frac{W}{9 \times 8} = \frac{W}{72}$.
The work rate of 1 woman is $\frac{1}{72}$ of the work per day.
Combined Group Rate
We need to find the combined work rate for 6 men and 3 women:
The work rate of 6 men is $6 \times \frac{1}{72} = \frac{6}{72} = \frac{1}{12}$ of the work per day.
The work rate of 3 women is $3 \times \frac{1}{72} = \frac{3}{72} = \frac{1}{24}$ of the work per day.
The combined work rate is the sum of their individual rates:
Combined rate = (Rate of 6 men) + (Rate of 3 women)
Combined rate = $\frac{1}{12} + \frac{1}{24} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}$ of the work per day.
Time to Complete Work
The time required to complete the work is calculated as:
Time = $\frac{\text{Total Work}}{\text{Combined Work Rate}}$
Time = $\frac{W}{\frac{1}{8} W} = \frac{1}{\frac{1}{8}} = 8$ days.
Thus, 3 women and 6 men can complete this piece of work in 8 days.
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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?
