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One man can do as much work in one day as a woman can do in $2$ days. A child does one third the work in a day as a woman. If an estate-owner hires $39$ pairs of hands, men, women and children in the ratio $6: 5: 2$ and pays them in all Rs. $1113$ at the end of the days work. What must the daily wages of a child be, if the wages are proportional to the amount of work done?

1. Rs. $14$
2. Rs. $5$
3. Rs. $20$
4. Rs. $7$

Let the amount of work done in a day by,

• Men $= 6 \frac{units}{day}$
• Women $= 3 \frac{units}{day}$
• Child $= 1 \frac{unit}{day}$

Each person has a piar of hands. So, $39$ pairs of hands $39$ peoples.

The ratio of Men:Women:Child $= 6:5:2$

• The number of Men $= \frac{6}{13} \times 39 = 18$
• The number of Women $= \frac{5}{13} \times 39 = 15$
• The number of Children $= \frac{2}{13} \times 39 = 6$

Now, work done by,

• Men $= 108$ units
• Women $= 45$ units
• Child $= 6$ units

Total work done $= 159$ units

So, the wages of $6$ Children $= \frac{1113}{159} \times 6 = ₹42$

$\therefore$ The daily wages of a child $=\frac{42}{6} =₹7.$

Correct Answer $: \text{D}$

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