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?
22
Solving Work and Time Problems with Multiple Groups
This problem involves calculating the time required for a group of men and women to complete a specific amount of work, given the time taken by two different groups. We can solve this type of problem by first determining the individual work rates of men and women.
Defining Work Rates
Let's assume:
- The work rate of one man is \(M\) units of work per day.
- The work rate of one woman is \(W\) units of work per day.
The total amount of work is constant. We can represent the total work as the product of the number of workers, their work rate, and the number of days.
Setting Up Equations from Given Information
We are given two scenarios:
- Two men and 7 women can complete the work in 28 days.
- Six men and 16 women can complete the same work in 11 days.
From the first scenario, the total work done by 2 men and 7 women in one day is \(2M + 7W\). Over 28 days, the total work done is \((2M + 7W) \times 28\). Let's assume the total work is 1 unit.
\[(2M + 7W) \times 28 = 1 \quad \small\text{(Equation 1)}\]
From the second scenario, the total work done by 6 men and 16 women in one day is \(6M + 16W\). Over 11 days, the total work done is \((6M + 16W) \times 11\).
\[(6M + 16W) \times 11 = 1 \quad \small\text{(Equation 2)}\]
Solving for Individual Work Rates
Now we have a system of two linear equations:
\[56M + 196W = 1 \quad \small\text{(from Equation 1)}\]
\[66M + 176W = 1 \quad \small\text{(from Equation 2)}\]
Since both expressions equal 1 (the total work), we can equate them:
\[56M + 196W = 66M + 176W\]
Rearrange the terms to group M and W:
\[196W - 176W = 66M - 56M\]
\[20W = 10M\]
Divide both sides by 10 to find the relationship between M and W:
\[2W = M\]
This tells us that the work rate of one man is equal to the work rate of two women.
Now, substitute \(M = 2W\) into either Equation 1 or Equation 2 to find the value of W. Let's use Equation 1:
\[(2(2W) + 7W) \times 28 = 1\]
\[(4W + 7W) \times 28 = 1\]
\[11W \times 28 = 1\]
\[308W = 1\]
So, the work rate of one woman is:
\[W = \frac{1}{308}\]
The work rate of one man is:
\[M = 2W = 2 \times \frac{1}{308} = \frac{2}{308} = \frac{1}{154}\]
Calculating Work Rate for the New Group
We need to find out how many days it will take for 5 men and 4 women to complete the same work. First, let's calculate their combined work rate per day:
\[\text{Combined work rate} = 5M + 4W\]
Substitute the values of M and W:
\[\text{Combined work rate} = 5 \times \frac{1}{154} + 4 \times \frac{1}{308}\]
To add these fractions, find a common denominator, which is 308:
\[\text{Combined work rate} = \frac{5 \times 2}{154 \times 2} + \frac{4}{308} = \frac{10}{308} + \frac{4}{308}\]
\[\text{Combined work rate} = \frac{10 + 4}{308} = \frac{14}{308}\]
Simplify the fraction:
\[\frac{14}{308} = \frac{14 \div 14}{308 \div 14} = \frac{1}{22}\]
So, 5 men and 4 women together complete \(\frac{1}{22}\) of the total work per day.
Calculating Time Taken
If the combined work rate is \(\frac{1}{22}\) units of work per day, and the total work is 1 unit, the number of days required to complete the work is the reciprocal of the work rate:
\[\text{Time} = \frac{\text{Total Work}}{\text{Combined Work Rate}} = \frac{1}{\frac{1}{22}} = 1 \times 22 = 22 \text{ days}\]
Therefore, 5 men and 4 women, working together, will complete the same work in 22 days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula Example |
|---|---|---|
| Work Rate | Amount of work done by a person or group in a unit of time (e.g., per day). | Rate = \( \frac{\text{Total Work}}{\text{Time Taken}} \) |
| Total Work | The entire task to be completed. Can often be assumed as 1 unit or the LCM of days. | Total Work = Rate \(\times\) Time |
| Efficiency | Often synonymous with work rate; higher efficiency means more work done per unit time. | Efficiency \(\propto\) Work Rate |
| Men-Days / Women-Days | A measure of total work unit. For example, 1 man-day is the work done by 1 man in 1 day. | Total Work = Number of Men \(\times\) Days \(\times\) Man's Rate Total Work = Number of Women \(\times\) Days \(\times\) Woman's Rate |
Additional Information: Alternative Methods
While the algebraic method using work rates \(M\) and \(W\) is systematic, another common approach for Work and Time problems, especially in competitive exams, is using the concept of "Total Work" as the Least Common Multiple (LCM) of the days given. However, when different groups of men and women are involved, the algebraic method to find the relationship between the work rates of men and women (like \(M=2W\) in this case) is often more straightforward.
Once the relationship (\(M=2W\)) is found, you can convert everyone into equivalent units of one type (e.g., women):
- Scenario 1: 2 men + 7 women = \(2(2W) + 7W = 4W + 7W = 11W\). This group (\(11W\)) completes the work in 28 days.
- Scenario 2: 6 men + 16 women = \(6(2W) + 16W = 12W + 16W = 28W\). This group (\(28W\)) completes the work in 11 days.
Notice that \(11W \times 28 \text{ days} = 308W \text{ work units}\). And \(28W \times 11 \text{ days} = 308W \text{ work units}\). So the total work is equivalent to 308 units of 'woman-days'.
Now, the new group is 5 men + 4 women. Convert this to women units:
- 5 men + 4 women = \(5(2W) + 4W = 10W + 4W = 14W\).
This group (\(14W\)) needs to complete 308 units of work. Time taken = Total Work / Work Rate.
\[\text{Time} = \frac{308W}{14W} = \frac{308}{14}\]
\[\text{Time} = 22 \text{ days}\]
This confirms the result obtained earlier and provides an alternative perspective using equivalent units.
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?
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?
40 men can complete a work in 15 days. Three days after they started working, 20 more men joined them. In how many days the total work will be completed?
