If 7 men and 4 women working together can fo five times as much work per hour as one man and one woman together then the ratio of the work done by a man and that of woman for a given time is :

Options

A. a) `2:3`
B. b) `3:2`
C. c) `2:1`
D. d) `1:2`

Answer

To solve the problem, we need to establish the relationship between the work done by men and women based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We have 7 men and 4 women working together.
- They can do 5 times as much work per hour as one man and one woman together.

2. **Setting Up the Equation:**
- Let the work done by one man in one hour be represented as \( m \).
- Let the work done by one woman in one hour be represented as \( w \).
- Therefore, the total work done by 7 men and 4 women in one hour is:
\[ 7m + 4w \]
- The work done by one man and one woman together in one hour is:
\[ m + w \]
- According to the problem, we have:
\[ 7m + 4w = 5(m + w) \]

3. **Expanding the Equation:**
- Expanding the right side:
\[ 7m + 4w = 5m + 5w \]

4. **Rearranging the Equation:**
- Now, let's rearrange the equation by moving all terms involving \( m \) to one side and all terms involving \( w \) to the other side:
\[ 7m - 5m = 5w - 4w \]
- This simplifies to:
\[ 2m = 1w \]

5. **Finding the Ratio:**
- From the equation \( 2m = 1w \), we can express the ratio of work done by men to work done by women:
\[ \frac{m}{w} = \frac{1}{2} \]
- Therefore, the ratio of the work done by a man to that of a woman is:
\[ 1 : 2 \]

### Final Answer:
The ratio of the work done by a man and that of a woman for a given time is \( 1 : 2 \).