Two men and 7 children complete a certain piece of work in 4 days while 4 men and 4 children complete the same work in only 3 days. The number of days required by 1 man to complete the work is

Options

A. 60 days
B. 15 days
C. 6 days
D. 51 days

Answer

To solve the problem, we need to determine how many days it takes for one man to complete a certain piece of work based on the information given about men and children working together.

### Step-by-Step Solution:

1. **Define Variables:**
- Let \( m \) be the amount of work done by one man in one day.
- Let \( c \) be the amount of work done by one child in one day.

2. **Work Done by Two Men and Seven Children:**
- The work done by two men in one day is \( 2m \).
- The work done by seven children in one day is \( 7c \).
- Therefore, the total work done by two men and seven children in one day is:
\[ 2m + 7c \]
- Since they complete the work in 4 days, the total work \( W \) can be expressed as:
\[ W = 4(2m + 7c) = 8m + 28c \]

3. **Work Done by Four Men and Four Children:**
- The work done by four men in one day is \( 4m \).
- The work done by four children in one day is \( 4c \).
- Therefore, the total work done by four men and four children in one day is:
\[ 4m + 4c \]
- Since they complete the work in 3 days, the total work \( W \) can also be expressed as:
\[ W = 3(4m + 4c) = 12m + 12c \]

4. **Setting the Two Expressions for Total Work Equal:**
- Since both expressions represent the same total work \( W \), we can set them equal to each other:
\[ 8m + 28c = 12m + 12c \]

5. **Rearranging the Equation:**
- Rearranging gives:
\[ 28c - 12c = 12m - 8m \]
\[ 16c = 4m \]
- Dividing both sides by 4:
\[ 4c = m \]
- This means that one man does the work of four children in one day.

6. **Finding Total Work in Terms of One Man's Work:**
- We can substitute \( m = 4c \) back into one of the expressions for \( W \). Using \( W = 8m + 28c \):
\[ W = 8(4c) + 28c = 32c + 28c = 60c \]

7. **Finding the Number of Days for One Man to Complete the Work:**
- If one man does \( m \) work in one day, then the number of days \( x \) required by one man to complete the work \( W \) is given by:
\[ W = x \cdot m \]
- Substituting \( W = 60c \) and \( m = 4c \):
\[ 60c = x \cdot 4c \]
- Dividing both sides by \( 4c \):
\[ x = \frac{60c}{4c} = 15 \]

### Conclusion:
The number of days required by one man to complete the work is **15 days**.