A man, a woman and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist 1 man and 1 woman to complete the job in `(1)/(4)` of a day?

Options

A. 1
B. 4
C. 19
D. 41

Answer

To solve the problem, we need to determine how many boys must assist one man and one woman to complete a job in \( \frac{1}{4} \) of a day.

### Step-by-Step Solution:

1. **Determine the Work Done by Each Individual in One Day:**
- A man can complete the job in 3 days, so his work rate is:
\[ \text{Work by 1 man in 1 day} = \frac{1}{3} \text{ of the job} \]
- A woman can complete the job in 4 days, so her work rate is:
\[ \text{Work by 1 woman in 1 day} = \frac{1}{4} \text{ of the job} \]
- A boy can complete the job in 12 days, so his work rate is:
\[ \text{Work by 1 boy in 1 day} = \frac{1}{12} \text{ of the job} \]

2. **Calculate the Total Work Done by 1 Man and 1 Woman in \( \frac{1}{4} \) of a Day:**
- Work done by 1 man in \( \frac{1}{4} \) of a day:
\[ \text{Work by 1 man in } \frac{1}{4} \text{ day} = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \]
- Work done by 1 woman in \( \frac{1}{4} \) of a day:
\[ \text{Work by 1 woman in } \frac{1}{4} \text{ day} = \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \]

3. **Combine the Work Done by the Man and Woman:**
- Total work done by 1 man and 1 woman in \( \frac{1}{4} \) of a day:
\[ \text{Total work} = \frac{1}{12} + \frac{1}{16} \]
- To add these fractions, find a common denominator (which is 48):
\[ \frac{1}{12} = \frac{4}{48}, \quad \frac{1}{16} = \frac{3}{48} \]
- Therefore,
\[ \text{Total work} = \frac{4}{48} + \frac{3}{48} = \frac{7}{48} \]

4. **Set Up the Equation for Boys' Contribution:**
- Let \( x \) be the number of boys. The work done by \( x \) boys in \( \frac{1}{4} \) of a day is:
\[ \text{Work by } x \text{ boys} = x \times \left(\frac{1}{12} \times \frac{1}{4}\right) = x \times \frac{1}{48} \]
- The total work done by the man, woman, and boys should equal 1 job:
\[ \frac{7}{48} + x \times \frac{1}{48} = 1 \]

5. **Solve for \( x \):**
- Rearranging the equation gives:
\[ x \times \frac{1}{48} = 1 - \frac{7}{48} \]
\[ x \times \frac{1}{48} = \frac{48 - 7}{48} = \frac{41}{48} \]
- Multiplying both sides by 48:
\[ x = 41 \]

### Final Answer:
Thus, **41 boys** must assist 1 man and 1 woman to complete the job in \( \frac{1}{4} \) of a day.

---