A can do a work in 6 days. B takes 8 days to complete it. C takes as long as A…

If a worker completes a job in t days, their work rate is 1/t of the job per day. When two people work together, their combined rate is the sum of their individual rates, and the time they take together is the reciprocal of that combined rate.

1. A completes the work in 6 days, so A's rate = 1/6 of the work per day.

2. B completes the work in 8 days, so B's rate = 1/8 of the work per day.

3. C takes as long as A and B would take working together. A and B's combined rate is 1/6 + 1/8 = 4/24 + 3/24 = 7/24 of the work per day, so their combined time is 24/7 days -- this is C's own time, which means C's own rate is also 7/24 of the work per day.

4. Adding B's rate and C's rate gives the rate at which B and C work together: 1/8 + 7/24 = 3/24 + 7/24 = 10/24 = 5/12 of the work per day.

5. The time B and C take together is the reciprocal of their combined rate: 12/5 days, that is, 2 2/5 days.

Using total work = 24 units (the LCM of 6 and 8): A does 4 units per day and B does 3 units per day, so together they do 7 units per day and finish in 24/7 days -- matching C's stated time, so C does 24 divided by (24/7) = 7 units per day. B and C together then do 3 + 7 = 10 units per day, finishing the 24 units in 24/10 = 2 2/5 days, confirming the rate-based answer.