In how manu days can the work be done by 9 men and 15 women ?
(i) 6 men and 5 women can complete the cork in 6 days.
(ii) 3 men and 4 women can complete the work in 10 days
(iii) 18 men and 15 women can complete the work in 2 days.

Options

A. (iii) only
B. All (i), (ii) and (iii)
C. Any two
D. Any one

Answer

To solve the problem of how many days it will take for 9 men and 15 women to complete the work, we will analyze the given statements step by step.

### Step 1: Understand the Work Done by Men and Women
We need to find the work efficiency of men and women based on the information provided in the statements. Let's denote the work done by one man in one day as \( M \) and the work done by one woman in one day as \( W \).

### Step 2: Analyze Statement (i)
From statement (i), we know:
- 6 men and 5 women can complete the work in 6 days.

The total work done can be expressed as:
\[ \text{Total Work} = \text{(Number of Workers)} \times \text{(Days)} = (6M + 5W) \times 6 \]
So, the total work is:
\[ \text{Total Work} = 36M + 30W \]

### Step 3: Analyze Statement (ii)
From statement (ii), we know:
- 3 men and 4 women can complete the work in 10 days.

The total work done can be expressed as:
\[ \text{Total Work} = (3M + 4W) \times 10 \]
So, the total work is:
\[ \text{Total Work} = 30M + 40W \]

### Step 4: Equate the Total Work from Statements (i) and (ii)
Since both expressions represent the same total work, we can set them equal to each other:
\[ 36M + 30W = 30M + 40W \]

### Step 5: Simplify the Equation
Rearranging the equation gives:
\[ 36M - 30M = 40W - 30W \]
\[ 6M = 10W \]
From this, we can derive the relationship:
\[ M = \frac{10}{6}W = \frac{5}{3}W \]

### Step 6: Analyze Statement (iii)
From statement (iii), we know:
- 18 men and 15 women can complete the work in 2 days.

The total work done can be expressed as:
\[ \text{Total Work} = (18M + 15W) \times 2 \]
So, the total work is:
\[ \text{Total Work} = 36M + 30W \]

### Step 7: Use the Relationship from Step 5
Now, we can substitute \( M \) in terms of \( W \) into the total work expression from statement (iii):
\[ 36\left(\frac{5}{3}W\right) + 30W = 60W + 30W = 90W \]
Thus, the total work is \( 90W \).

### Step 8: Calculate the Work Done by 9 Men and 15 Women
Now, we need to find out how many days it will take for 9 men and 15 women to complete the same total work of \( 90W \):
\[ \text{Work done by 9 men and 15 women in one day} = 9M + 15W \]
Substituting \( M = \frac{5}{3}W \):
\[ 9\left(\frac{5}{3}W\right) + 15W = 15W + 15W = 30W \]

### Step 9: Calculate the Number of Days
Now, we can find the number of days \( D \) it takes for 9 men and 15 women to complete the work:
\[ D = \frac{\text{Total Work}}{\text{Work done in one day}} = \frac{90W}{30W} = 3 \text{ days} \]

### Conclusion
Therefore, the work can be completed by 9 men and 15 women in **3 days**.

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