Three men can complete a work in 6 days and five women can complete the same work in 18 days. In how many days can 4 men and 10 women together complete the same work?
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Solving Work and Time Problems with Men and Women
This problem involves calculating the time taken by a group of men and women working together to complete a task, given the time taken by individual groups of men and women.
To solve this, we first need to determine the individual work rate of one man and one woman per day. The work rate is the fraction of the total work completed in one day.
Calculating Individual Work Rates
We are given that:
- 3 men can complete the work in 6 days.
- 5 women can complete the same work in 18 days.
Let the total work be 1 unit.
Work Rate of Men:
3 men complete the work in 6 days.
Work done by 3 men in 1 day is $\frac{1}{6}$ of the total work.
Therefore, the work done by 1 man in 1 day is $\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}$ of the total work.
So, 1 man's 1-day work rate is $\frac{1}{18}$.
Work Rate of Women:
5 women complete the work in 18 days.
Work done by 5 women in 1 day is $\frac{1}{18}$ of the total work.
Therefore, the work done by 1 woman in 1 day is $\frac{1}{18} \times \frac{1}{5} = \frac{1}{90}$ of the total work.
So, 1 woman's 1-day work rate is $\frac{1}{90}$.
Calculating Combined Work Rate (4 Men and 10 Women)
We need to find how many days 4 men and 10 women together can complete the same work. First, let's find their combined work rate per day.
Work done by 4 men in 1 day = $4 \times (\text{1 man's 1-day work rate}) = 4 \times \frac{1}{18} = \frac{4}{18} = \frac{2}{9}$ of the work.
Work done by 10 women in 1 day = $10 \times (\text{1 woman's 1-day work rate}) = 10 \times \frac{1}{90} = \frac{10}{90} = \frac{1}{9}$ of the work.
Combined work done by 4 men and 10 women in 1 day = (Work by 4 men in 1 day) + (Work by 10 women in 1 day)
Combined work rate per day = $\frac{2}{9} + \frac{1}{9} = \frac{2+1}{9} = \frac{3}{9} = \frac{1}{3}$ of the work.
So, 4 men and 10 women together complete $\frac{1}{3}$ of the work in 1 day.
Calculating Total Days to Complete the Work
If 4 men and 10 women complete $\frac{1}{3}$ of the work in 1 day, the total number of days required to complete the entire work (which is 1 unit of work) is the reciprocal of their combined daily work rate.
Total days = $\frac{\text{Total Work}}{\text{Combined Work Rate per Day}} = \frac{1}{\frac{1}{3}} = 1 \times 3 = 3$ days.
Therefore, 4 men and 10 women together can complete the work in 3 days.
Summary of Calculations
| Group | Time to Complete Work (Days) | Total Work Units (in 1 day) | Work Rate per Person per Day |
|---|---|---|---|
| 3 Men | 6 | $\frac{1}{6}$ | $\frac{1}{6} \times \frac{1}{3} = \frac{1}{18}$ (for 1 man) |
| 5 Women | 18 | $\frac{1}{18}$ | $\frac{1}{18} \times \frac{1}{5} = \frac{1}{90}$ (for 1 woman) |
| 4 Men | - | $4 \times \frac{1}{18} = \frac{2}{9}$ | - |
| 10 Women | - | $10 \times \frac{1}{90} = \frac{1}{9}$ | - |
| 4 Men + 10 Women | ? | $\frac{2}{9} + \frac{1}{9} = \frac{3}{9} = \frac{1}{3}$ | - |
Time taken by 4 men and 10 women = $\frac{1}{\text{Combined daily work rate}} = \frac{1}{\frac{1}{3}} = 3$ days.
Revision Table: Key Work and Time Concepts
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Work Rate | The amount of work done per unit of time (e.g., per day). | Work Rate = $\frac{\text{Total Work}}{\text{Time Taken}}$ |
| Time Taken | The total time required to complete a task. | Time Taken = $\frac{\text{Total Work}}{\text{Work Rate}}$ |
| Total Work | The entire task to be completed (often considered as 1 unit). | Total Work = Work Rate $\times$ Time Taken |
| Combined Work Rate | The sum of individual work rates when multiple people or entities work together. | Combined Rate = Rate$_1$ + Rate$_2$ + ... |
Additional Information: Variations in Work and Time Problems
Work and time problems often involve different scenarios:
- Individual vs. Combined Work: Calculating rates of individuals or groups working together.
- Changing Workloads: Problems where the amount of work is different.
- Varying Efficiency: Cases where efficiency changes, affecting the work rate.
- Alternating Work: Problems where individuals work on alternate days or shifts.
- Pipes and Cisterns: Similar concept applied to filling or emptying tanks using pipes (inlet and outlet pipes).
Understanding the concept of work rate as the reciprocal of the time taken is fundamental to solving these problems. If someone completes a work in 'N' days, their 1-day work rate is $\frac{1}{N}$. When people work together, their daily work rates add up.
Important Questions from Time and Work
A man, a woman, and a boy can do a work in 6, 9, and 18 days respectively. How many boys must assist one man and one woman to do the work in 1 day?
A, B, and C worked together for 5 days to build a wall and then B left the work. A and C together finished the remaining work in 5 days. In how many days can A and C working together finish building the whole wall if B alone can do it in 25 days?
Mother, daughter, and son can do a work in 3, 4, and 5 days respectively. How many days will they take to complete the work, if they work together?
Rajan does 7/11 of a work in 21 days. How many more days will he take to complete the work?
By looking in a mirror, it appears that it is 6:30 in the clock. What is the real time?
