A,B and C can complete a job in 20 days, 30 days, and 40 days, respectively. They worked for 6 days and then A left. B and C completed the job. Find B's wages in the total wages of Rs. 20,000 paid to them ( in Rs.)

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C, and then calculate B's share of the total wages.

### Step-by-Step Solution:

1. **Determine the work rates of A, B, and C:**
- A can complete the job in 20 days, so A's work rate is \( \frac{1}{20} \) of the job per day.
- B can complete the job in 30 days, so B's work rate is \( \frac{1}{30} \) of the job per day.
- C can complete the job in 40 days, so C's work rate is \( \frac{1}{40} \) of the job per day.

2. **Calculate the combined work rate of A, B, and C:**
\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{30} + \frac{1}{40} \]
To add these fractions, we need a common denominator. The least common multiple of 20, 30, and 40 is 120.
\[ \frac{1}{20} = \frac{6}{120}, \quad \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
\[ \text{Combined work rate} = \frac{6 + 4 + 3}{120} = \frac{13}{120} \]

3. **Calculate the work done by A, B, and C in 6 days:**
\[ \text{Work done in 6 days} = 6 \times \frac{13}{120} = \frac{78}{120} = \frac{13}{20} \]
This means that after 6 days, \( \frac{13}{20} \) of the job is completed.

4. **Calculate the remaining work:**
\[ \text{Remaining work} = 1 - \frac{13}{20} = \frac{7}{20} \]

5. **Determine the work rate of B and C together:**
\[ \text{B's work rate} = \frac{1}{30}, \quad \text{C's work rate} = \frac{1}{40} \]
\[ \text{Combined work rate of B and C} = \frac{1}{30} + \frac{1}{40} \]
Using a common denominator of 120:
\[ \frac{1}{30} = \frac{4}{120}, \quad \frac{1}{40} = \frac{3}{120} \]
\[ \text{Combined work rate of B and C} = \frac{4 + 3}{120} = \frac{7}{120} \]

6. **Calculate the time taken by B and C to complete the remaining work:**
Let \( x \) be the number of days B and C work together to finish the remaining work.
\[ \text{Work done by B and C in } x \text{ days} = x \times \frac{7}{120} \]
Setting this equal to the remaining work:
\[ x \times \frac{7}{120} = \frac{7}{20} \]
To solve for \( x \):
\[ x = \frac{7}{20} \times \frac{120}{7} = 6 \text{ days} \]

7. **Total days worked by each person:**
- A worked for 6 days.
- B worked for \( 6 + 6 = 12 \) days.
- C worked for \( 6 + 6 = 12 \) days.

8. **Calculate the total work done by each person:**
- Work done by A:
\[ 6 \times \frac{1}{20} = \frac{6}{20} = \frac{3}{10} \]
- Work done by B:
\[ 12 \times \frac{1}{30} = \frac{12}{30} = \frac{2}{5} = \frac{8}{20} \]
- Work done by C:
\[ 12 \times \frac{1}{40} = \frac{12}{40} = \frac{3}{10} \]

9. **Calculate the total work done:**
\[ \text{Total work} = \frac{3}{10} + \frac{8}{20} + \frac{3}{10} = \frac{6}{20} + \frac{8}{20} + \frac{6}{20} = \frac{20}{20} = 1 \]

10. **Calculate the ratio of work done:**
- A's share: \( \frac{3}{10} \)
- B's share: \( \frac{8}{20} = \frac{2}{5} \)
- C's share: \( \frac{3}{10} \)

The ratio of work done by A, B, and C is:
\[ 3:4:3 \]

11. **Calculate B's wages from the total wages of Rs. 20,000:**
- Total parts = \( 3 + 4 + 3 = 10 \)
- B's share = \( 4 \) parts
\[ \text{B's wages} = \frac{4}{10} \times 20000 = 8000 \text{ Rs.} \]

### Final Answer:
B's wages are Rs. 8,000.