The ratio of time taken by a man, a woman, and a child to complete a work individually is $1 : 2 : 3$ respectively.

If $10$ men, $15$ women, and $18$ children work together and receive a total wage of Rs. $1410$, what will be the total wage for $10$ men, $12$ women, and $15$ children working together?

Question

The ratio of time taken by a man, a woman, and a child to complete a work individually is $1 : 2 : 3$ respectively.

If $10$ men, $15$ women, and $18$ children work together and receive a total wage of Rs. $1410$, what will be the total wage for $10$ men, $12$ women, and $15$ children working together?

Accepted Answer

**Solution:**

1. Let the time taken by a man to complete the work be $t$ units. Then, the time taken by a woman is $2t$ and by a child is $3t$.

2. The work done per unit time (rate of work) is inversely proportional to the time taken. So,
   - Work rate of a man = $\frac{1}{t}$
   - Work rate of a woman = $\frac{1}{2t}$
   - Work rate of a child = $\frac{1}{3t}$

3. Calculate the total work done per unit time by $10$ men, $15$ women, and $18$ children:
   $$10 	imes \frac{1}{t} + 15 	imes \frac{1}{2t} + 18 	imes \frac{1}{3t} = \frac{10}{t} + \frac{15}{2t} + \frac{18}{3t}$$

4. Simplify the expression:
   $$= \frac{10}{t} + \frac{15}{2t} + \frac{6}{t} = \frac{10}{t} + \frac{7.5}{t} + \frac{6}{t} = \frac{23.5}{t}$$

5. The total wage Rs. $1410$ is shared in the ratio of work done by each group. Since total work rate is $\frac{23.5}{t}$, the wage per unit work rate is:
   $$	ext{Wage per unit work rate} = \frac{1410}{23.5/t} = 1410 	imes \frac{t}{23.5}$$

6. Now, calculate the total work rate for $10$ men, $12$ women, and $15$ children:
   $$10 	imes \frac{1}{t} + 12 	imes \frac{1}{2t} + 15 	imes \frac{1}{3t} = \frac{10}{t} + \frac{6}{t} + \frac{5}{t} = \frac{21}{t}$$

7. The total wage for this group is:
   $$	ext{Wage} = 	ext{Wage per unit work rate} 	imes 	ext{Total work rate} = 1410 	imes \frac{t}{23.5} 	imes \frac{21}{t} = 1410 	imes \frac{21}{23.5}$$

8. Calculate the value:
   $$1410 	imes \frac{21}{23.5} = 1410 	imes 0.8936 \approx 1260$$

**Answer:** Rs. 1260

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