To solve the problem step by step, we need to determine how much should be paid to 'B' based on the work done by 'A', 'B', and 'C' together.
### Step 1: Determine the work done by 'A' and 'B'
- 'A' can complete the work in 10 days, so in one day, 'A' does \( \frac{1}{10} \) of the work.
- 'B' can complete the work in 15 days, so in one day, 'B' does \( \frac{1}{15} \) of the work.
### Step 2: Calculate the total work done by 'A' and 'B' in one day
- The combined work done by 'A' and 'B' in one day is:
\[
\text{Work by A} + \text{Work by B} = \frac{1}{10} + \frac{1}{15}
\]
- To add these fractions, we need a common denominator. The least common multiple of 10 and 15 is 30.
\[
\frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30}
\]
- Therefore:
\[
\text{Total work in one day} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}
\]
### Step 3: Calculate the total work done by 'A', 'B', and 'C' in 5 days
- Together, 'A', 'B', and 'C' completed the work in 5 days. Thus, the total work done in 5 days is:
\[
\text{Total work} = 5 \times \left(\text{Work done by A, B, and C in one day}\right)
\]
- We know that the total work done in one day by 'A' and 'B' is \( \frac{1}{6} \). Let the work done by 'C' in one day be \( C \).
- Therefore, the total work done in one day by 'A', 'B', and 'C' is:
\[
\frac{1}{6} + C
\]
- In 5 days, they complete the entire work, which is 1 (the whole work). Thus:
\[
5 \left(\frac{1}{6} + C\right) = 1
\]
\[
\frac{5}{6} + 5C = 1
\]
\[
5C = 1 - \frac{5}{6} = \frac{1}{6}
\]
\[
C = \frac{1}{30}
\]
### Step 4: Calculate the efficiency of 'A', 'B', and 'C'
- Now we know:
- 'A' does \( \frac{1}{10} \) of the work in one day.
- 'B' does \( \frac{1}{15} \) of the work in one day.
- 'C' does \( \frac{1}{30} \) of the work in one day.
### Step 5: Calculate the ratio of work done
- The efficiencies are:
- Efficiency of 'A' = 3 (since \( \frac{1}{10} = \frac{3}{30} \))
- Efficiency of 'B' = 2 (since \( \frac{1}{15} = \frac{2}{30} \))
- Efficiency of 'C' = 1 (since \( \frac{1}{30} = \frac{1}{30} \))
- The ratio of work done by 'A', 'B', and 'C' is 3:2:1.
### Step 6: Calculate the share of 'B'
- The total ratio parts = \( 3 + 2 + 1 = 6 \).
- The total payment is Rs. 42000.
- 'B's share is:
\[
\text{B's share} = \frac{2}{6} \times 42000 = \frac{1}{3} \times 42000 = 14000
\]
### Final Answer
Thus, 'B' should be paid Rs. 14,000.
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