Step 1: Understanding the Question:
The problem asks us to determine the respective daily wages of three workers: A, B, and C.
We are given the individual times A and B take to complete a task, which are 10 days and 15 days respectively.
A and B work together for exactly 5 days before C joins to finish the remaining work in 2 days.
The total wage allocated for completing the entire piece of work is Rs. 3000.
Step 2: Key Formula or Approach:
Wages in such problems are distributed in direct proportion to the actual amount of work done by each individual, not necessarily the total time they spent.
We will use the LCM (Least Common Multiple) method to define the total units of work.
Once we find the per-day work rate (efficiency) for each person, we can calculate their total work contribution and thus their total wage.
Dividing the total wage of each person by the number of days they worked gives their daily wage.
Step 3: Detailed Explanation:
Let the total amount of work be the LCM of the days taken by A and B, which is $LCM(10, 15) = 30$ units.
The efficiency (work done per day) of worker A is $\frac{30}{10} = 3$ units per day.
The efficiency of worker B is $\frac{30}{15} = 2$ units per day.
A and B work together for 5 days.
The total work done by A and B in these 5 days is $5 \times (3 + 2) = 5 \times 5 = 25$ units.
The remaining work to be completed is $30 - 25 = 5$ units.
Worker C finishes these remaining 5 units of work in 2 days.
Therefore, the efficiency of C is $\frac{5}{2} = 2.5$ units per day.
The total wage of Rs. 3000 is paid for completing the total 30 units of work.
Thus, the wage paid per single unit of work is $\frac{3000}{30} = 100$ Rs. per unit.
Now we calculate the daily wage for each person, which equals their daily work rate multiplied by the wage per unit.
The daily wage of worker A is $3 \text{ units/day} \times 100 \text{ Rs./unit} = 300$ Rs.
The daily wage of worker B is $2 \text{ units/day} \times 100 \text{ Rs./unit} = 200$ Rs.
The daily wage of worker C is $2.5 \text{ units/day} \times 100 \text{ Rs./unit} = 250$ Rs.
The respective daily wages of A, B, and C are 300, 200, and 250.
Step 4: Final Answer:
The daily wages for A, B, and C respectively are Rs. 300, Rs. 200, and Rs. 250.