A can do a piece of work in 15 days, B can do the same work in 10 days, and C do the same work in 12 days. All three of them do the same work together, then they collectively get Rs. 9000. If B’s share is divided among three new persons D, E and F in the ratio of 1 : 5 : 3 respectively then find the share of F.

Solution

Given:
– A can complete the work in 15 days.
– B can complete the work in 10 days.
– C can complete the work in 12 days.
– The total payment for the work is Rs. 9000.

Step 1: Calculate the total work done by A, B, and C

Let the total work be \(W\) units.
– A’s work rate: \(\frac{W}{15}\) units/day
– B’s work rate: \(\frac{W}{10}\) units/day
– C’s work rate: \(\frac{W}{12}\) units/day

Step 2: Calculate the total work done by A, B, and C together in one day

\[ \text{Total work rate} = \frac{W}{15} + \frac{W}{10} + \frac{W}{12} = \frac{4W}{60} + \frac{6W}{60} + \frac{5W}{60} = \frac{15W}{60} = \frac{W}{4} \text{ units/day} \]

Step 3: Calculate the share of B

Since the total payment is for 1 day of work, B’s share is proportional to his work rate:
\[ \text{B’s share} = \frac{\text{B’s work rate}}{\text{Total work rate}} \times \text{Total payment} = \frac{\frac{W}{10}}{\frac{W}{4}} \times 9000 = \frac{4}{10} \times 9000 = Rs. 3600 \]

Step 4: Calculate F’s share

B’s share is divided among D, E, and F in the ratio of 1:5:3. The total parts in the ratio are \(1 + 5 + 3 = 9\).
– F’s share = \(\frac{3}{9} \times \text{B’s share} = \frac{3}{9} \times 3600 = Rs. 1200\)

Conclusion

The share of F is Rs. 1200.