One man and six women working together can do a job in 10 days. The same job is done by two men in 'p' days and by eight women in `p + 5` days. By what percentage is the efficiency of a man greater than that of a woman?

Options

A. 300%
B. 500%
C. 600%
D. 700%

Answer

To solve the problem step by step, we will define the efficiencies of men and women, set up equations based on the information given, and then calculate the percentage difference in their efficiencies.

### Step 1: Define Variables
Let the efficiency of one man be \( m \) and the efficiency of one woman be \( w \).

### Step 2: Set Up the First Equation
According to the problem, one man and six women can complete the job in 10 days. Therefore, the total work done can be expressed as:
\[ \text{Total Work} = \text{(Efficiency of 1 man + Efficiency of 6 women)} \times \text{Time} \]
This gives us:
\[ 10(m + 6w) = \text{Total Work} \]

### Step 3: Set Up the Second Equation
The same job can be done by two men in \( p \) days, which gives us:
\[ p(2m) = \text{Total Work} \]

The job can also be done by eight women in \( p + 5 \) days, giving us:
\[ (p + 5)(8w) = \text{Total Work} \]

### Step 4: Equate the Total Work
Since all three expressions represent the same total work, we can set them equal to each other:
\[ 10(m + 6w) = 2mp = 8w(p + 5) \]

### Step 5: Simplify the Equations
From the first equation:
\[ 10m + 60w = 2mp \quad \text{(1)} \]
From the second equation:
\[ 10m + 60w = 8wp + 40w \quad \text{(2)} \]

### Step 6: Rearranging the Equations
From equation (1):
\[ 2mp = 10m + 60w \quad \Rightarrow \quad p = \frac{10m + 60w}{2m} \]

From equation (2):
\[ 8wp + 40w = 10m + 60w \quad \Rightarrow \quad 8wp = 10m + 20w \quad \Rightarrow \quad p = \frac{10m + 20w}{8w} \]

### Step 7: Set the Two Expressions for \( p \) Equal
Now we have two expressions for \( p \):
\[ \frac{10m + 60w}{2m} = \frac{10m + 20w}{8w} \]

### Step 8: Cross Multiply and Solve for \( w/m \)
Cross multiplying gives:
\[ 8w(10m + 60w) = 2m(10m + 20w) \]
Expanding both sides:
\[ 80mw + 480w^2 = 20m^2 + 40mw \]
Rearranging gives:
\[ 40mw + 480w^2 - 20m^2 = 0 \]

### Step 9: Factor the Equation
This can be factored or solved using the quadratic formula, but for simplicity, we can find the ratio of \( w/m \) directly from the coefficients.

### Step 10: Find the Ratio of Efficiencies
From the previous steps, we can conclude that:
\[ \frac{w}{m} = \frac{1}{6} \]
This means that the efficiency of a man is 6 times that of a woman.

### Step 11: Calculate the Percentage Difference
To find the percentage by which the efficiency of a man is greater than that of a woman:
\[ \text{Percentage Difference} = \left(\frac{m - w}{w}\right) \times 100 \]
Substituting \( m = 6w \):
\[ \text{Percentage Difference} = \left(\frac{6w - w}{w}\right) \times 100 = \left(\frac{5w}{w}\right) \times 100 = 500\% \]

### Final Answer
The efficiency of a man is greater than that of a woman by **500%**.