X' men can complete a work in 4 days. '(X-2)' women can complete the same work in 8 days, while ratio of efficiency of man to woman is 3 : 2. If `'(x)/(2)'` men & '(X - 2)' women started the work & worked for only 2 days then, how many women should replace to men to complete the remaining work in three days?

Options

A. 2 women
B. 8 women
C. 6 women
D. 4 women

Answer

To solve the problem step by step, we will break down the information given and calculate accordingly.

### Step 1: Calculate Total Work Done by Men and Women
Given:
- \( X \) men can complete the work in 4 days.
- \( (X - 2) \) women can complete the same work in 8 days.
- The efficiency ratio of men to women is \( 3:2 \).

Let the efficiency of one man be \( M = 3k \) and the efficiency of one woman be \( W = 2k \).

**Total work done by men:**
\[ \text{Total Work} = \text{Number of Men} \times \text{Efficiency of Men} \times \text{Time} \]
\[ = X \times 3k \times 4 = 12Xk \]

**Total work done by women:**
\[ \text{Total Work} = \text{Number of Women} \times \text{Efficiency of Women} \times \text{Time} \]
\[ = (X - 2) \times 2k \times 8 = 16(X - 2)k \]

Since both expressions represent the same total work, we can set them equal:
\[ 12Xk = 16(X - 2)k \]

### Step 2: Simplify the Equation
Dividing both sides by \( k \) (assuming \( k \neq 0 \)):
\[ 12X = 16(X - 2) \]
Expanding the right side:
\[ 12X = 16X - 32 \]
Rearranging gives:
\[ 32 = 16X - 12X \]
\[ 32 = 4X \]
\[ X = 8 \]

### Step 3: Calculate the Number of Men and Women
Now substituting \( X \) back:
- Number of Men = \( X = 8 \)
- Number of Women = \( X - 2 = 6 \)

### Step 4: Work Done in 2 Days
Now, \( \frac{X}{2} = \frac{8}{2} = 4 \) men and \( (X - 2) = 6 \) women start the work.

**Daily work done:**
\[ \text{Work by Men in 1 day} = 4 \times 3 = 12 \]
\[ \text{Work by Women in 1 day} = 6 \times 2 = 12 \]
\[ \text{Total Work in 1 day} = 12 + 12 = 24 \]

**Work done in 2 days:**
\[ \text{Total Work in 2 days} = 24 \times 2 = 48 \]

### Step 5: Calculate Remaining Work
**Total Work:**
\[ \text{Total Work} = 12Xk = 12 \times 8k = 96k \]

**Remaining Work:**
\[ \text{Remaining Work} = 96k - 48 = 48k \]

### Step 6: Work Required in 3 Days
To complete the remaining work in 3 days:
\[ \text{Work required per day} = \frac{48k}{3} = 16k \]

### Step 7: Determine How Many Women are Needed
Let \( Y \) be the number of women replacing men. The efficiency of \( Y \) women is:
\[ \text{Work done by Y women in 1 day} = Y \times 2k \]

Setting the total work done in a day to 16k:
\[ Y \times 2k = 16k \]
Dividing both sides by \( 2k \):
\[ Y = 8 \]

### Step 8: Calculate the Number of Women to Replace
Initially, there were 6 women. To find how many women need to be added:
\[ \text{Women to replace} = Y - 6 = 8 - 6 = 2 \]

### Final Answer
**The number of women that should replace men to complete the remaining work in three days is 2.**

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