If 12 men or 20 women can do a work in 54 days, then in how many days can 9 men and 12 women together do the work?

Options

A. 38
B. 32
C. 40
D. 35

Answer

To solve the problem step by step, we will follow the logic presented in the video transcript.

### Step 1: Determine the work done by men and women
We know that:
- 12 men can complete the work in 54 days.
- 20 women can complete the same work in 54 days.

We can express the total work in terms of men and women.

**Total work (in man-days) = Number of men × Time taken by men**
\[ \text{Total Work} = 12 \text{ men} \times 54 \text{ days} = 648 \text{ man-days} \]

**Total work (in woman-days) = Number of women × Time taken by women**
\[ \text{Total Work} = 20 \text{ women} \times 54 \text{ days} = 1080 \text{ woman-days} \]

### Step 2: Find the efficiency of men and women
From the above calculations, we can find the efficiency of men and women.

Let the efficiency of 1 man be \( m \) and the efficiency of 1 woman be \( w \).

From the total work done:
- The work done by 12 men in 54 days:
\[ 12m \times 54 = 648 \]
\[ 12m = \frac{648}{54} = 12 \]
\[ m = 1 \text{ man-day} \]

- The work done by 20 women in 54 days:
\[ 20w \times 54 = 1080 \]
\[ 20w = \frac{1080}{54} = 20 \]
\[ w = 1 \text{ woman-day} \]

### Step 3: Establish the efficiency ratio
From the calculations, we find that:
- Efficiency of 1 man = 1
- Efficiency of 1 woman = 0.6 (since \( 20w = 12m \) gives us \( w = \frac{3}{5}m \))

Thus, the efficiency ratio of men to women is:
\[ \frac{m}{w} = \frac{5}{3} \]

### Step 4: Calculate the combined efficiency of 9 men and 12 women
Now, we need to find the combined efficiency of 9 men and 12 women:
- Efficiency of 9 men:
\[ 9m = 9 \times 1 = 9 \text{ man-days} \]

- Efficiency of 12 women:
\[ 12w = 12 \times 0.6 = 7.2 \text{ man-days} \]

**Total efficiency of 9 men and 12 women:**
\[ \text{Total Efficiency} = 9 + 7.2 = 16.2 \text{ man-days} \]

### Step 5: Calculate the time taken to complete the work
Now, we can calculate the time taken to complete the work using the total work and the combined efficiency:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Total Efficiency}} \]
\[ \text{Time} = \frac{648 \text{ man-days}}{16.2 \text{ man-days}} \approx 40 \text{ days} \]

### Final Answer
Thus, 9 men and 12 women together can complete the work in **40 days**.

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