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 working together can complete a job in 30 days. The ratio of their efficiencies is 3 : 2. In how many days can the faster person complete the job?
A takes 15 days to complete \(\frac{5}{7} \) of a work. With the help of B, they finish the whole work in 12 days. In how many days, B alone will complete the same work
A alone can complete a work in 14 days and B alone can complete the same work in 21 days. A and B start the work together but A leaves the work after 4 days of the starting of work. In how many days B will complete the remaining work?
For completing a certain work, A is 50% less efficient than B and B is 50% more efficient than C. Working together A, B and C can complete the work in 48 days. A alone can complete the same work in:
30 persons can do a piece of work in 24 days. How many more persons are required to complete the work in 20 days?
