To solve the problem step by step, we will determine the efficiencies of A, B, and C, and then calculate how much each will receive based on their contributions to the work.
### Step 1: Determine the efficiencies of A and B
- A can complete the work in 16 days. Therefore, A's efficiency is:
\[
\text{Efficiency of A} = \frac{1}{16} \text{ (work per day)}
\]
- B can complete the work in 24 days. Therefore, B's efficiency is:
\[
\text{Efficiency of B} = \frac{1}{24} \text{ (work per day)}
\]
### Step 2: Determine the combined efficiency of A and B
- The combined efficiency of A and B is:
\[
\text{Combined Efficiency of A and B} = \frac{1}{16} + \frac{1}{24}
\]
- To add these fractions, we need a common denominator. The least common multiple (LCM) of 16 and 24 is 48.
\[
\frac{1}{16} = \frac{3}{48}, \quad \frac{1}{24} = \frac{2}{48}
\]
- Therefore,
\[
\text{Combined Efficiency of A and B} = \frac{3}{48} + \frac{2}{48} = \frac{5}{48}
\]
### Step 3: Determine the combined efficiency of A, B, and C
- Together, A, B, and C complete the work in 6 days. Thus, their combined efficiency is:
\[
\text{Combined Efficiency of A, B, and C} = \frac{1}{6}
\]
### Step 4: Find C's efficiency
- We know:
\[
\text{Efficiency of A, B, and C} = \text{Efficiency of A and B} + \text{Efficiency of C}
\]
- Substituting the known values:
\[
\frac{1}{6} = \frac{5}{48} + \text{Efficiency of C}
\]
- To find C's efficiency, we rearrange the equation:
\[
\text{Efficiency of C} = \frac{1}{6} - \frac{5}{48}
\]
- Finding a common denominator (which is 48):
\[
\frac{1}{6} = \frac{8}{48}
\]
- Therefore,
\[
\text{Efficiency of C} = \frac{8}{48} - \frac{5}{48} = \frac{3}{48} = \frac{1}{16}
\]
### Step 5: Calculate the total efficiency of A, B, and C
- The efficiencies are:
- A: \(\frac{3}{48}\)
- B: \(\frac{2}{48}\)
- C: \(\frac{3}{48}\)
### Step 6: Calculate the total parts of work done
- Total parts = Efficiency of A + Efficiency of B + Efficiency of C:
\[
\text{Total parts} = 3 + 2 + 3 = 8
\]
### Step 7: Calculate the remuneration for each
- Total remuneration = 400 rupees.
- The share of each person is calculated based on their parts of work:
- A's share:
\[
\text{A's share} = \frac{3}{8} \times 400 = 150 \text{ rupees}
\]
- B's share:
\[
\text{B's share} = \frac{2}{8} \times 400 = 100 \text{ rupees}
\]
- C's share:
\[
\text{C's share} = \frac{3}{8} \times 400 = 150 \text{ rupees}
\]
### Final Answer:
- A will receive 150 rupees, B will receive 100 rupees, and C will receive 150 rupees.
