$A$, $B$ and $C$ can do a ... $A$, $B$ and $C$ can do a piece of work in $30$, $40$ and $50$ days respectively. If $A$ and $B$ work in alternate days started by $A$ and they get the assistance of $C$ all the days, find in how many days the whole work will be finished?
My Attempt:
In $30$ days, $A$ does $1$ work.
In $1$ day, $A$ does $\frac {1}{30}$ work.
In $40$ days, $B$ does $1$ work.
In $1$ day, $B$ does $\frac {1}{40}$ work.
In $50$ days, $C$ does $1$ work.
In $1$ day, $C$ does $\frac {1}{50}$ work.
In $1$ day, $(A+C)$ do $\frac {4}{75}$ work.
In $1$ day, $(B+C)$ do $\frac {9}{200}$ work.
I could not solve from here. Please help.

A takes-> 30 days, B takes-> 40 days, C takes-> 50 days
Total unit of work they have to finish is(LCM of 30,40,50) i.e 600 units
It means 
A does -> 20 unit of work in 1 day, B does -> 15 unit in 1 day and C does 12 units in 1 day.
On 1st day A+C does 32 units out of 600, On 2nd day B+C does next 27 units of remaining work.
It means they do 59 unit of work in 2 days. In 20 days 590 units will be done by them. Remaining 10 units are done by A+C in \frac{10}{32}$$ days. 
Total days they took to finish the work 600/59 =10 \frac{5}{16}$$ = 20 \frac{5}{16}$$ days.
