6 men complete a piece of work in 12 days. 8 women can complete the same piece of work in 18 days. Whereas 18 children can complete the piece of work in 10 days. 4 men, 12 women and 20 children work together for 2 days, and then only 36 men were to complete the remaining work in x day.
56x soldiers can complete a piece of work in 24 days. In how many days can 42 soldiers complete the same piece of work?

Options

A. 32 days
B. 24 days
C. 16 days
D. 48 days

Answer

To solve the problem step by step, let's break it down into manageable parts.

### Step 1: Calculate the total work in terms of efficiency

1. **Men's Efficiency**:
- 6 men complete the work in 12 days.
- Total work = Efficiency × Time = 6m × 12 = 72m (where m is the efficiency of one man).

2. **Women's Efficiency**:
- 8 women complete the work in 18 days.
- Total work = 8w × 18 = 144w (where w is the efficiency of one woman).

3. **Children's Efficiency**:
- 18 children complete the work in 10 days.
- Total work = 18c × 10 = 180c (where c is the efficiency of one child).

Since the total work is the same, we can set these equations equal to each other:

\[ 72m = 144w = 180c \]

### Step 2: Establish the ratio of efficiencies

From the equations above, we can express the efficiencies in terms of a common variable:

1. From \( 72m = 144w \):
\[ m = 2w \]

2. From \( 72m = 180c \):
\[ m = \frac{5}{4}c \]

Now we can express everything in terms of \( c \):

- From \( m = \frac{5}{4}c \), we can substitute into \( m = 2w \):
\[ 2w = \frac{5}{4}c \]
\[ w = \frac{5}{8}c \]

Now we have:
- \( m = \frac{5}{4}c \)
- \( w = \frac{5}{8}c \)

### Step 3: Calculate the total work done by 4 men, 12 women, and 20 children in 2 days

1. **Work done by 4 men in 2 days**:
\[ \text{Work by 4 men} = 4m \times 2 = 8m \]

2. **Work done by 12 women in 2 days**:
\[ \text{Work by 12 women} = 12w \times 2 = 24w \]

3. **Work done by 20 children in 2 days**:
\[ \text{Work by 20 children} = 20c \times 2 = 40c \]

Now substituting \( m \) and \( w \) in terms of \( c \):

- Work by 4 men:
\[ 8m = 8 \times \frac{5}{4}c = 10c \]

- Work by 12 women:
\[ 24w = 24 \times \frac{5}{8}c = 15c \]

- Work by 20 children:
\[ 40c = 40c \]

Total work done in 2 days:
\[ \text{Total work} = 10c + 15c + 40c = 65c \]

### Step 4: Calculate remaining work

Total work (from earlier) is \( 720 \) (using \( 72m \)).

Remaining work after 2 days:
\[ \text{Remaining work} = 720 - 65c \]

### Step 5: Calculate the efficiency of 36 men

Efficiency of 36 men:
\[ \text{Efficiency} = 36m = 36 \times \frac{5}{4}c = 45c \]

### Step 6: Calculate the time \( x \) for 36 men to complete the remaining work

Using the formula:
\[ \text{Remaining work} = \text{Efficiency} \times \text{Time} \]
\[ 720 - 65c = 45c \times x \]

Rearranging gives:
\[ 720 - 65c = 45cx \]
\[ 720 = 45cx + 65c \]
\[ 720 = c(45x + 65) \]

### Step 7: Calculate the value of \( c \)

From the total work:
\[ 720 = 180c \]
\[ c = 4 \]

### Step 8: Substitute \( c \) back to find \( x \)

Substituting \( c \):
\[ 720 = 4(45x + 65) \]
\[ 720 = 180x + 260 \]
\[ 460 = 180x \]
\[ x = \frac{460}{180} = \frac{23}{9} \]

### Step 9: Calculate how many days 42 soldiers can complete the work

Given that 56 soldiers can complete the work in 24 days:
\[ \text{Total work} = 56 \times 24 = 1344 \]

Now, we need to find how many days \( d \) it takes for 42 soldiers to complete the same work:
\[ 1344 = 42d \]
\[ d = \frac{1344}{42} = 32 \]

### Final Answer
Thus, 42 soldiers can complete the work in **32 days**.

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