If 7 men and 4 women working together, can do 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 a woman for a given time is:

Options

A. `2:3`
B. `3:2`
C. `2:1`
D. `1:2`

Answer

To solve the problem, we need to find the ratio of the work done by a man (x) to that done by a woman (y) based on the information provided. Here are the steps:

### Step 1: Define the efficiencies
Let:
- The efficiency of one man be \( x \) (work done by one man in one hour).
- The efficiency of one woman be \( y \) (work done by one woman in one hour).

### Step 2: Write the equation for the work done
According to the problem, 7 men and 4 women working together can do five times the work done by one man and one woman together.

The total work done by 7 men and 4 women in one hour can be expressed as:
\[ 7x + 4y \]

The total work done by 1 man and 1 woman in one hour is:
\[ x + y \]

Since the work done by 7 men and 4 women is five times that of 1 man and 1 woman, we can write the equation:
\[ 7x + 4y = 5(x + y) \]

### Step 3: Simplify the equation
Expanding the right side of the equation gives:
\[ 7x + 4y = 5x + 5y \]

Now, rearranging the equation to isolate terms involving \( x \) and \( y \):
\[ 7x - 5x = 5y - 4y \]
This simplifies to:
\[ 2x = y \]

### Step 4: Find the ratio of work done
To find the ratio of the work done by a man to that done by a woman, we need to express \( x \) in terms of \( y \):
\[ \frac{x}{y} = \frac{1}{2} \]

### Step 5: State the final ratio
Thus, the ratio of the work done by a man to that done by a woman is:
\[ 1 : 2 \]

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

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