Understanding Work and Wage Ratios

The question deals with the concept of work done and corresponding wages for men, women, and children. The work done by them is in a specific ratio, which implies their wages for the same unit of time (like per week) are also in that same ratio, assuming they are paid based on their work output or potential. The ratio of work done (and thus wages per unit of time) by a man, a woman, and a child is given as 3 : 2 : 1.

Setting up the Wage Relationship

Let's represent the weekly wage of a child as a variable. Since the ratio of wages for a man, a woman, and a child is 3:2:1:

• Let the weekly wage of a child = $\text{Rs. } x$
• Then the weekly wage of a woman = $\text{Rs. } 2x$
• And the weekly wage of a man = $\text{Rs. } 3x$

Calculating Total Weekly Wages for the First Group

We are given that in the factory, there are 20 men, 30 women, and 36 children, and their total weekly wages amount to Rs. 78.

Let's calculate the total weekly wages based on our variables:

• Total weekly wages for 20 men = $20 \times (\text{weekly wage of a man}) = 20 \times (3x) = 60x$
• Total weekly wages for 30 women = $30 \times (\text{weekly wage of a woman}) = 30 \times (2x) = 60x$
• Total weekly wages for 36 children = $36 \times (\text{weekly wage of a child}) = 36 \times (x) = 36x$

The total weekly wages for this group is the sum of these amounts:

Total weekly wages = $60x + 60x + 36x = 156x$

Finding the Value of the Wage Unit (x)

We know the total weekly wages for the first group is Rs. 78. So, we can set up an equation:

$\text{156x} = 78$

Now, we can solve for $x$:

$x = \frac{78}{156}$

Simplifying the fraction:

$x = \frac{1}{2} = 0.5$

So, the weekly wage of a child is Rs. 0.5.

Determining Individual Weekly Wages

Using the value of $x$, we can find the actual weekly wages for each person:

• Weekly wage of a child = $x = \text{Rs. } 0.5$
• Weekly wage of a woman = $2x = 2 \times 0.5 = \text{Rs. } 1$
• Weekly wage of a man = $3x = 3 \times 0.5 = \text{Rs. } 1.5$

Calculating Weekly Wages for the Second Group

We need to find the wages for 15 men, 21 women, and 30 children for 2 weeks. First, let's calculate their total wages for one week:

• Weekly wages for 15 men = $15 \times (\text{weekly wage of a man}) = 15 \times 1.5 = \text{Rs. } 22.5$
• Weekly wages for 21 women = $21 \times (\text{weekly wage of a woman}) = 21 \times 1 = \text{Rs. } 21$
• Weekly wages for 30 children = $30 \times (\text{weekly wage of a child}) = 30 \times 0.5 = \text{Rs. } 15$

The total weekly wages for this second group is:

Total weekly wages for the second group = $22.5 + 21 + 15 = \text{Rs. } 58.5$

Calculating Total Wages for Two Weeks

The question asks for the wages for 2 weeks. So, we multiply the total weekly wages for the second group by 2:

Total wages for 2 weeks = (Total weekly wages for the second group) $\times$ 2

Total wages for 2 weeks = $58.5 \times 2 = \text{Rs. } 117$

Thus, the wages of 15 men, 21 women, and 30 children for 2 weeks will be Rs. 117.