A can do $\dfrac 12$ of a piece of work in $5$ days. B can do $3S$ of the same work in $9$ days and C can do $\dfrac 23$ of that work in $8$ days. In how many days can three of them together do the work.

Assuming the typo 3S means 3/5, A does half the work in 5 days, so A's full work rate is 1/10. B does 3/5 of the work in 9 days, so B's rate is 1/15. C does 2/3 of the work in 8 days, so C's rate is 1/12. Using the sum of rates formula, their combined rate is 1/10 + 1/15 + 1/12 = 1/4 of the work per day. The total time to complete the work together is the reciprocal of their combined rate, which is 4 days.

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