12 men can do a work in 10 days while 10 women can do the same work in 18 days. In how many days 4 men & 6 women together can do the same work?

Options

A. `(120)/(7)` days
B. 24 days
C. `(180)/(13)` days
D. 15 days

Answer

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

### Step 1: Determine the total work done by men and women
We know that:
- 12 men can complete the work in 10 days.
- 10 women can complete the same work in 18 days.

Let's denote the efficiency of one man as \( M \) and the efficiency of one woman as \( W \).

The total work can be calculated using the formula:
\[ \text{Total Work} = \text{Number of Workers} \times \text{Efficiency} \times \text{Time} \]

For men:
\[ \text{Total Work} = 12M \times 10 = 120M \]

For women:
\[ \text{Total Work} = 10W \times 18 = 180W \]

Since the total work is the same in both cases, we can set these equations equal to each other:
\[ 120M = 180W \]

### Step 2: Find the ratio of the efficiencies of men and women
We can simplify the equation to find the ratio of \( M \) to \( W \):
\[ \frac{M}{W} = \frac{180}{120} = \frac{3}{2} \]

This means that the efficiency of men is 1.5 times that of women.

### Step 3: Express the efficiency of men and women in terms of a common variable
Let’s express the efficiency of men and women in terms of a common variable:
- Let \( W = 2x \) (efficiency of one woman)
- Then, \( M = 3x \) (efficiency of one man)

### Step 4: Calculate the total efficiency of 4 men and 6 women
Now, we can calculate the total efficiency of 4 men and 6 women:
\[ \text{Total Efficiency} = 4M + 6W \]
Substituting the values of \( M \) and \( W \):
\[ \text{Total Efficiency} = 4(3x) + 6(2x) = 12x + 12x = 24x \]

### Step 5: Calculate the total work in terms of \( x \)
Now we can calculate the total work using either men or women’s efficiency. Let’s use the total work calculated from men:
\[ \text{Total Work} = 120M = 120(3x) = 360x \]

### Step 6: Find the number of days required for 4 men and 6 women to complete the work
Now, we can find the number of days required for 4 men and 6 women to complete the work:
\[ \text{Days} = \frac{\text{Total Work}}{\text{Total Efficiency}} = \frac{360x}{24x} \]

The \( x \) cancels out:
\[ \text{Days} = \frac{360}{24} = 15 \]

Thus, 4 men and 6 women together can complete the work in **15 days**.

### Final Answer:
The answer is **15 days**.