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 estateowner hires 39 pairs of hands, men, women and children in the ratio 6:5:2 and pays them in all Rs. 1,113 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?

Options

A. Rs. 14
B. Rs. 5
C. Rs. 20
D. Rs. 7

Answer

To solve the problem step by step, we will break down the information given and calculate the daily wages of a child based on the work done by men, women, and children.

### Step 1: Define the variables
Let the number of men, women, and children be represented in the ratio 6:5:2. We can express this as:
- Number of men = 6x
- Number of women = 5x
- Number of children = 2x

### Step 2: Set up the equation for total workers
According to the problem, the total number of workers is 39. Therefore, we can set up the equation:
\[ 6x + 5x + 2x = 39 \]

### Step 3: Solve for x
Combine the terms:
\[ 13x = 39 \]
Now, divide both sides by 13:
\[ x = 3 \]

### Step 4: Calculate the number of men, women, and children
Now, substitute x back into the expressions for men, women, and children:
- Number of men = \( 6x = 6 \times 3 = 18 \)
- Number of women = \( 5x = 5 \times 3 = 15 \)
- Number of children = \( 2x = 2 \times 3 = 6 \)

### Step 5: Determine the work done by each group
From the problem, we know:
- A man can do as much work in one day as a woman can do in two days. Thus, if we let the work done by a man in one day be W, then:
- Work done by a woman in one day = \( \frac{W}{2} \)

- A child does one-third the work of a woman in one day. Therefore:
- Work done by a child in one day = \( \frac{1}{3} \times \frac{W}{2} = \frac{W}{6} \)

### Step 6: Calculate total work done
Now we can calculate the total work done by all the workers in one day:
- Total work done by men = Number of men × Work done by a man
\[ = 18 \times W = 18W \]

- Total work done by women = Number of women × Work done by a woman
\[ = 15 \times \frac{W}{2} = \frac{15W}{2} \]

- Total work done by children = Number of children × Work done by a child
\[ = 6 \times \frac{W}{6} = W \]

Now, add all the work done:
\[ \text{Total work} = 18W + \frac{15W}{2} + W \]

To combine these, convert \( W \) and \( 18W \) into fractions with a common denominator:
\[ 18W = \frac{36W}{2} \]
\[ W = \frac{2W}{2} \]

So,
\[ \text{Total work} = \frac{36W}{2} + \frac{15W}{2} + \frac{2W}{2} = \frac{53W}{2} \]

### Step 7: Calculate the wages
The total payment for all workers is Rs. 1,113. Since the wages are proportional to the work done, we can find the wage per unit of work:
Let the wage per unit of work be \( P \):
\[ P \times \frac{53W}{2} = 1113 \]
\[ P = \frac{1113 \times 2}{53W} \]

### Step 8: Calculate the daily wages of a child
The daily wage of a child is:
\[ \text{Wage of a child} = \text{Work done by a child} \times P \]
\[ = \frac{W}{6} \times P \]

Substituting the value of \( P \):
\[ = \frac{W}{6} \times \frac{1113 \times 2}{53W} \]
\[ = \frac{1113 \times 2}{6 \times 53} \]
\[ = \frac{2226}{318} \]
\[ = 7 \]

Thus, the daily wages of a child must be Rs. 7.