4 men and 2 boys can finish a piece of work in 5 days. 3 women and 4 boys can finish the same work in 5 days. Also 2 men and 3 women can finish the same work in 5 days. In how many days 1 man, 1 woman and one boy can finish the work, at their double efficiency?

Options

A. `4(8)/(13)`
B. `4(4)/(13)`
C. `3(7)/(13)`
D. none of these

Answer

To solve the problem step by step, we will denote the efficiencies of one man, one woman, and one boy as \( m \), \( w \), and \( b \) respectively.

### Step 1: Set up the equations based on the work done
From the problem, we have three scenarios:

1. **4 men and 2 boys finish the work in 5 days:**
\[ 4m + 2b = \frac{1}{5} \quad \text{(since they complete 1 unit of work in 5 days)} \]

2. **3 women and 4 boys finish the work in 5 days:**
\[ 3w + 4b = \frac{1}{5} \]

3. **2 men and 3 women finish the work in 5 days:**
\[ 2m + 3w = \frac{1}{5} \]

### Step 2: Multiply each equation by 5 to eliminate the fractions
1. \( 20m + 10b = 1 \) (Equation 1)
2. \( 15w + 20b = 1 \) (Equation 2)
3. \( 10m + 15w = 1 \) (Equation 3)

### Step 3: Solve the equations
We will solve these equations step by step.

**From Equation 1:**
\[ 20m + 10b = 1 \implies 2m + b = \frac{1}{10} \quad \text{(Divide by 10)} \]
**From Equation 2:**
\[ 15w + 20b = 1 \implies 3w + 4b = \frac{1}{5} \quad \text{(Already in this form)} \]
**From Equation 3:**
\[ 10m + 15w = 1 \implies 2m + 3w = \frac{1}{5} \quad \text{(Divide by 5)} \]

Now we have:
1. \( 2m + b = \frac{1}{10} \) (Equation 4)
2. \( 3w + 4b = \frac{1}{5} \) (Equation 2)
3. \( 2m + 3w = \frac{1}{5} \) (Equation 3)

### Step 4: Substitute to find values
From Equation 4, express \( b \):
\[ b = \frac{1}{10} - 2m \]

Substituting \( b \) into Equation 2:
\[ 3w + 4\left(\frac{1}{10} - 2m\right) = \frac{1}{5} \]
\[ 3w + \frac{4}{10} - 8m = \frac{2}{10} \]
\[ 3w - 8m = \frac{2}{10} - \frac{4}{10} = -\frac{2}{10} \implies 3w - 8m = -\frac{1}{5} \quad \text{(Equation 5)} \]

Now substitute \( b \) into Equation 3:
\[ 2m + 3w = \frac{1}{5} \]

### Step 5: Solve the system of equations (Equations 5 and 3)
Now we have two equations:
1. \( 3w - 8m = -\frac{1}{5} \)
2. \( 2m + 3w = \frac{1}{5} \)

From Equation 2, express \( w \):
\[ 3w = \frac{1}{5} - 2m \implies w = \frac{1}{15} - \frac{2m}{3} \]

Substituting \( w \) into Equation 5:
\[ 3\left(\frac{1}{15} - \frac{2m}{3}\right) - 8m = -\frac{1}{5} \]
\[ \frac{1}{5} - 2m - 8m = -\frac{1}{5} \]
\[ -10m = -\frac{2}{5} \implies m = \frac{1}{25} \]

Now substitute \( m \) back to find \( w \) and \( b \):
\[ w = \frac{1}{15} - \frac{2 \cdot \frac{1}{25}}{3} = \frac{1}{15} - \frac{2}{75} = \frac{5}{75} - \frac{2}{75} = \frac{3}{75} = \frac{1}{25} \]
\[ b = \frac{1}{10} - 2 \cdot \frac{1}{25} = \frac{1}{10} - \frac{2}{25} = \frac{5}{50} - \frac{4}{50} = \frac{1}{50} \]

### Step 6: Calculate the combined efficiency of 1 man, 1 woman, and 1 boy
Now we have:
- \( m = \frac{1}{25} \)
- \( w = \frac{1}{25} \)
- \( b = \frac{1}{50} \)

Combined efficiency:
\[ m + w + b = \frac{1}{25} + \frac{1}{25} + \frac{1}{50} = \frac{2}{25} + \frac{1}{50} = \frac{4}{50} + \frac{1}{50} = \frac{5}{50} = \frac{1}{10} \]

### Step 7: Calculate the time taken at double efficiency
At double efficiency:
\[ 2(m + w + b) = 2 \cdot \frac{1}{10} = \frac{1}{5} \]

Total work = 1 unit, so time taken:
\[ \text{Time} = \frac{\text{Total Work}}{\text{New Efficiency}} = \frac{1}{\frac{1}{5}} = 5 \text{ days} \]

### Final Answer
Thus, 1 man, 1 woman, and 1 boy can finish the work in **5 days** at their double efficiency.

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