Four men and 6 women can complete a certain piece of work in 5 days whereas three men and 4 women can complete it in 7 days. How many men should assist 25 women to complete times the same work in 5 days?

Options

A. 10
B. 4
C. 8
D. 5

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Define the Efficiency of Men and Women
Let the efficiency of one man be \( x \) and the efficiency of one woman be \( y \).

### Step 2: Set Up the Equations
From the problem statement, we have two scenarios:

1. Four men and six women can complete the work in 5 days:
\[ \text{Work} = \text{Efficiency} \times \text{Time} \]
\[ 5 \times (4x + 6y) = \text{Total Work} \quad \text{(Equation 1)} \]

2. Three men and four women can complete the work in 7 days:
\[ 7 \times (3x + 4y) = \text{Total Work} \quad \text{(Equation 2)} \]

### Step 3: Equate the Total Work
Since both equations represent the same total work, we can set them equal to each other:
\[ 5(4x + 6y) = 7(3x + 4y) \]

### Step 4: Expand and Simplify
Expanding both sides gives:
\[ 20x + 30y = 21x + 28y \]

Rearranging the equation:
\[ 20x - 21x + 30y - 28y = 0 \]
\[ -x + 2y = 0 \quad \Rightarrow \quad x = 2y \]

### Step 5: Substitute to Find Total Work
Now, substitute \( x = 2y \) back into one of the original equations to find the total work. Using Equation 1:
\[ \text{Total Work} = 5(4(2y) + 6y) = 5(8y + 6y) = 5(14y) = 70y \]

### Step 6: Determine the Work with 25 Women
Now we need to find how many men \( n \) are required to assist 25 women to complete 2.5 times the same work in 5 days:
\[ \text{Total Work for 2.5 times} = 2.5 \times 70y = 175y \]

### Step 7: Set Up the New Equation
The equation for the work done by \( n \) men and 25 women in 5 days is:
\[ 5(n \cdot x + 25 \cdot y) = 175y \]
Substituting \( x = 2y \):
\[ 5(n \cdot 2y + 25y) = 175y \]
\[ 5(2ny + 25y) = 175y \]
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ 5(2n + 25) = 175 \]

### Step 8: Solve for \( n \)
Dividing both sides by 5:
\[ 2n + 25 = 35 \]
Subtracting 25 from both sides:
\[ 2n = 10 \]
Dividing by 2:
\[ n = 5 \]

### Final Answer
The number of men required to assist 25 women to complete the work in 5 days is \( \boxed{5} \).

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