40 men and 20 women together can complete a work in 12 days and ratio of efficiency of a man to a woman is 2:3, find how many men are required to complete half of the work in 7 days?

Options

A. A)70
B. B)35
C. C)30
D. D)60

Answer

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

### Step 1: Calculate Total Work Done
We know that 40 men and 20 women can complete the work in 12 days.

- The efficiency of a man is given as 2 units.
- The efficiency of a woman is given as 3 units.

**Total Efficiency Calculation:**
- Efficiency of 40 men = 40 men × 2 units/man = 80 units
- Efficiency of 20 women = 20 women × 3 units/woman = 60 units

**Total Efficiency of the group:**
\[ \text{Total Efficiency} = 80 + 60 = 140 \text{ units/day} \]

**Total Work Calculation:**
Total work can be calculated as:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} \]
\[ \text{Total Work} = 140 \text{ units/day} \times 12 \text{ days} = 1680 \text{ units} \]

### Step 2: Calculate Half of the Total Work
Since we need to find how many men are required to complete half of the work:
\[ \text{Half of Total Work} = \frac{1680}{2} = 840 \text{ units} \]

### Step 3: Calculate the Required Efficiency to Complete Half Work in 7 Days
We need to find out how many men (let's denote this as \( m \)) are required to complete 840 units of work in 7 days.

**Required Efficiency Calculation:**
\[ \text{Required Efficiency} = \frac{\text{Half of Total Work}}{\text{Time}} \]
\[ \text{Required Efficiency} = \frac{840 \text{ units}}{7 \text{ days}} = 120 \text{ units/day} \]

### Step 4: Relate the Required Efficiency to the Number of Men
The efficiency of \( m \) men is:
\[ \text{Efficiency of } m \text{ men} = m \times 2 \text{ units/man} \]
Setting this equal to the required efficiency:
\[ m \times 2 = 120 \]

### Step 5: Solve for \( m \)
\[ m = \frac{120}{2} = 60 \]

Thus, **the number of men required to complete half of the work in 7 days is 60.**

### Final Answer:
**60 men are required.**

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