12 men can complete a piece of work in 4 days, while 15 women can complete the same work in 4 days. 6 men start working on the job and after working for two days, all of them stopped working. How many women should be put on the job to complete the remaining work, if It is to be completed in 3 days

Options

A. 15
B. 22
C. 18
D. Data inadequate

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the total work done by men and women
We know that:
- 12 men can complete the work in 4 days.
- 15 women can also complete the same work in 4 days.

First, we calculate the total work in terms of man-days and woman-days.

Total work done by men:
\[ \text{Total Work} = \text{Number of Men} \times \text{Days} = 12 \text{ men} \times 4 \text{ days} = 48 \text{ man-days} \]

Total work done by women:
\[ \text{Total Work} = \text{Number of Women} \times \text{Days} = 15 \text{ women} \times 4 \text{ days} = 60 \text{ woman-days} \]

### Step 2: Equate the total work done by men and women
Since both complete the same work:
\[ 48 \text{ man-days} = 60 \text{ woman-days} \]

From this, we can find the efficiency ratio of men to women:
\[ \frac{M}{W} = \frac{60}{48} = \frac{5}{4} \]
This means that 5 man-days are equivalent to 4 woman-days.

### Step 3: Calculate the work done by 6 men in 2 days
Now, we need to find out how much work 6 men can do in 2 days.

The efficiency of 1 man is:
\[ \text{Efficiency of 1 man} = \frac{48 \text{ man-days}}{12 \text{ men} \times 4 \text{ days}} = 1 \text{ unit of work per day} \]

Thus, the work done by 6 men in 1 day:
\[ \text{Work done by 6 men in 1 day} = 6 \text{ men} \times 1 \text{ unit} = 6 \text{ units} \]

In 2 days, the work done will be:
\[ \text{Total work done by 6 men in 2 days} = 6 \text{ units/day} \times 2 \text{ days} = 12 \text{ units} \]

### Step 4: Calculate the remaining work
The total work is 48 units (from Step 1). The remaining work after 6 men have worked for 2 days is:
\[ \text{Remaining Work} = 48 \text{ units} - 12 \text{ units} = 36 \text{ units} \]

### Step 5: Determine how many women are needed to complete the remaining work in 3 days
Let \( x \) be the number of women needed to complete the remaining 36 units of work in 3 days.

The work done by \( x \) women in 3 days is:
\[ \text{Work done by } x \text{ women in 3 days} = x \text{ women} \times 3 \text{ days} \times \frac{4 \text{ units}}{1 \text{ woman}} = 12x \text{ units} \]

Setting this equal to the remaining work:
\[ 12x = 36 \]

### Step 6: Solve for \( x \)
Dividing both sides by 12:
\[ x = \frac{36}{12} = 3 \]

### Conclusion
Thus, **3 women** are needed to complete the remaining work in 3 days.

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