If 9 men and 12 boys can do a piece of work in 4 days and 4 men and 16 boys can do the same piece of work in 6 days, how long will 6 men and 24 boys take to complete the same work?
4 days
Solving Work and Time Problems: Men and Boys Efficiency
This problem involves understanding the combined work rate of different groups of workers (men and boys) to determine the time taken to complete a piece of work.
Let's denote the work rate of one man as \(M\) units of work per day and the work rate of one boy as \(B\) units of work per day.
Setting Up Equations for Work Done
The total amount of work done is the same in all cases. The formula for work done is: $$\text{Work} = \text{Rate} \times \text{Time}$$ where Rate is the combined rate of all workers.
From the problem statement, we have two scenarios:
- 9 men and 12 boys can do the work in 4 days.
- 4 men and 16 boys can do the same work in 6 days.
The combined rate for the first group is \( (9M + 12B) \). The work done is \( (9M + 12B) \times 4 \).
The combined rate for the second group is \( (4M + 16B) \). The work done is \( (4M + 16B) \times 6 \).
Since the work is the same in both cases, we can set up the equation:
$$ (9M + 12B) \times 4 = (4M + 16B) \times 6 $$Finding the Relationship Between Men's and Boys' Work Rates
Let's simplify the equation to find the relationship between \(M\) and \(B\):
$$ 36M + 48B = 24M + 96B $$Subtract \(24M\) from both sides:
$$ 36M - 24M + 48B = 96B $$ $$ 12M + 48B = 96B $$Subtract \(48B\) from both sides:
$$ 12M = 96B - 48B $$ $$ 12M = 48B $$Divide both sides by 12:
$$ M = 4B $$This tells us that the work rate of one man is equal to the work rate of four boys.
Calculating the Total Work
Now that we know \(M = 4B\), we can calculate the total work using either of the initial scenarios.
Using the first scenario (9 men and 12 boys in 4 days):
Replace \(M\) with \(4B\):
Combined rate = \( 9M + 12B = 9(4B) + 12B = 36B + 12B = 48B \)
Total Work = Rate \(\times\) Time = \( 48B \times 4 = 192B \)
Alternatively, using the second scenario (4 men and 16 boys in 6 days):
Replace \(M\) with \(4B\):
Combined rate = \( 4M + 16B = 4(4B) + 16B = 16B + 16B = 32B \)
Total Work = Rate \(\times\) Time = \( 32B \times 6 = 192B \)
Both calculations give the same total work, which is \(192B\) (in terms of boys' work rate) or \(192B = 192 \times (M/4) = 48M\) (in terms of men's work rate). Let's stick with \(192B\).
Calculating the Time for 6 Men and 24 Boys
We need to find out how long it will take 6 men and 24 boys to complete the same work.
First, find the combined work rate of 6 men and 24 boys. Replace \(M\) with \(4B\):
Combined rate = \( 6M + 24B = 6(4B) + 24B = 24B + 24B = 48B \)
Now, calculate the time taken:
Time = Total Work / Combined Rate
Time = \( \frac{192B}{48B} \)
Time = \( \frac{192}{48} \)
Time = \( 4 \) days
Summary of Calculation
Let's summarize the key steps and results:
| Step | Description | Result |
|---|---|---|
| 1 | Equate total work from two scenarios | \( (9M + 12B) \times 4 = (4M + 16B) \times 6 \) |
| 2 | Find relationship between M and B | \( M = 4B \) |
| 3 | Calculate total work (using \( M=4B \)) | \( (9(4B) + 12B) \times 4 = 192B \) |
| 4 | Calculate combined rate of 6 men & 24 boys (using \( M=4B \)) | \( 6(4B) + 24B = 48B \) |
| 5 | Calculate time taken | \( \text{Time} = \frac{\text{Total Work}}{\text{Combined Rate}} = \frac{192B}{48B} \) |
| 6 | Final Answer | 4 days |
Therefore, 6 men and 24 boys will take 4 days to complete the same work.
Revision Table: Work and Time Concepts
| Concept | Explanation |
|---|---|
| Work Rate | The amount of work done by a person or group per unit of time. E.g., work/day or work/hour. |
| Total Work | The total amount of task to be completed. It is often treated as '1 unit' or a calculated value like in this problem (192B). |
| Relationship between Work, Rate, Time | Work = Rate × Time. This is the fundamental formula for these types of problems. |
| Combined Rate | When multiple individuals or groups work together, their rates are added up to find the combined rate. |
| Efficiency | Often related to work rate. Higher efficiency means a higher work rate. In this problem, a man is 4 times as efficient as a boy (\(M=4B\)). |
Additional Information: Solving Work Problems
Work and time problems are common in quantitative aptitude. They often involve finding the efficiency of individuals or groups and then using that information to calculate the time taken for a different group or individual to complete the same or a different amount of work.
- Always define variables clearly for the work rate of each type of worker (e.g., M for man, B for boy).
- Set up equations based on the given scenarios using the formula Work = Rate × Time.
- If the total work is the same in different scenarios, equate the expressions for work done.
- Solve the equations to find the relationship between the work rates of different workers.
- Calculate the total amount of work based on one of the given scenarios and the work rate relationship.
- Calculate the combined work rate of the new group.
- Use the formula Time = Total Work / Combined Rate to find the required time.
- Ensure units are consistent throughout the calculation (e.g., days, hours).
This problem highlights how understanding the relative efficiency of different workers is key to solving such problems.
Important Questions from Work Efficiency
A and B together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
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 can 7 men complete the same work?
