3 men can do a piece of worki n 18 days , 6 boys can also do the same work in 18 days. In how many days, 4 men and 4 boys together will finish the work?

Options

A. 10days
B. 6 days
C. 21 days
D. 9 days

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Determine the work done by men and boys
We know that:
- 3 men can complete the work in 18 days.
- 6 boys can also complete the same work in 18 days.

Let’s denote:
- The work done by 1 man in 1 day as \( m \) (units of work).
- The work done by 1 boy in 1 day as \( b \) (units of work).

### Step 2: Calculate total work in terms of men and boys
The total work done by 3 men in 18 days can be expressed as:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} \times \text{Work per Man per Day} = 3 \times 18 \times m = 54m \]

Similarly, the total work done by 6 boys in 18 days is:
\[ \text{Total Work} = \text{Number of Boys} \times \text{Days} \times \text{Work per Boy per Day} = 6 \times 18 \times b = 108b \]

Since both expressions represent the same total work, we can set them equal to each other:
\[ 54m = 108b \]

### Step 3: Find the ratio of work done by men and boys
Dividing both sides by 54 gives:
\[ m = 2b \]

This means the efficiency of 1 man is equivalent to the efficiency of 2 boys.

### Step 4: Calculate the efficiency of 4 men and 4 boys
Now, we need to find the combined efficiency of 4 men and 4 boys:
- The work done by 4 men in one day is:
\[ 4m = 4 \times 2b = 8b \]
- The work done by 4 boys in one day is:
\[ 4b \]

Thus, the total work done by 4 men and 4 boys together in one day is:
\[ \text{Total Efficiency} = 8b + 4b = 12b \]

### Step 5: Calculate the number of days required to complete the work
Now we need to find out how many days \( d \) it will take for 4 men and 4 boys to finish the total work \( 108b \):
\[ \text{Total Work} = \text{Efficiency} \times \text{Days} \]
\[ 108b = 12b \times d \]

Dividing both sides by \( 12b \) gives:
\[ d = \frac{108b}{12b} = 9 \]

### Conclusion
Thus, 4 men and 4 boys together will finish the work in **9 days**.