4 men or 6 women or 10 children can paint a house in 5 days. The painting is given to a couple and their 5 sons. They finish the job in

Options

A. `11/60 days `
B. `5 (5)/(11) days `
C. `5 (6)/(11) days `
D. `11 (1)/(5) days `

Answer

To solve the problem, we need to determine how long it takes for a couple and their 5 sons to paint a house, given the information about the work capacity of men, women, and children.

### Step-by-Step Solution:

1. **Understand the Work Capacity**:
We know that:
- 4 men can paint a house in 5 days.
- 6 women can paint a house in 5 days.
- 10 children can paint a house in 5 days.

2. **Calculate Work Done by Each Group**:
The total work done by each group in terms of "house" per day can be calculated:
- Work done by 4 men in 5 days = 1 house
- Therefore, work done by 1 man in 1 day = 1/20 houses.
- Work done by 6 women in 5 days = 1 house
- Therefore, work done by 1 woman in 1 day = 1/30 houses.
- Work done by 10 children in 5 days = 1 house
- Therefore, work done by 1 child in 1 day = 1/50 houses.

3. **Determine the Work Capacity of the Couple and Their Sons**:
Let's denote:
- 1 man = 1/20 houses/day
- 1 woman = 1/30 houses/day
- 1 child = 1/50 houses/day

The couple consists of 1 man and 1 woman, and they have 5 sons (children). The work done by the couple and their 5 sons in one day is:
\[ \text{Work done by couple} = \left(\frac{1}{20} + \frac{1}{30}\right) + 5 \times \frac{1}{50} \]

4. **Calculate the Work Done by the Couple**:
First, we need to find a common denominator to add the fractions:
- The least common multiple of 20, 30, and 50 is 300.
- Convert each fraction:
- \(\frac{1}{20} = \frac{15}{300}\)
- \(\frac{1}{30} = \frac{10}{300}\)
- \(\frac{1}{50} = \frac{6}{300}\)

Now, substituting these values into the equation:
\[ \text{Work done by couple} = \left(\frac{15}{300} + \frac{10}{300}\right) + 5 \times \frac{6}{300} \]
\[ = \frac{25}{300} + \frac{30}{300} = \frac{55}{300} \]
\[ = \frac{11}{60} \text{ houses per day} \]

5. **Calculate the Time Taken to Paint the House**:
If the couple and their 5 sons can paint \(\frac{11}{60}\) of a house in one day, the time taken to paint one whole house is:
\[ \text{Time} = \frac{1 \text{ house}}{\frac{11}{60} \text{ houses/day}} = \frac{60}{11} \text{ days} \]
Which can be approximated as:
\[ = 5 \frac{5}{11} \text{ days} \]

### Final Answer:
The couple and their 5 sons finish the job in \( \frac{60}{11} \) days, or approximately \( 5 \frac{5}{11} \) days.

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