$A$, $B$ and $C$ together can do a piece of work in $8\mathit{}$ days. $A$ alone can do it in $20$ days, $A$ and $B$ can do it in $14$ days. If $A$ and $C$ can do the same work in $\frac{k}{29}$ days, then find the value of $k$.

$A$ and $B$ together can do a piece of work in $20$ days. $B$ alone can do $\frac{1}{5}\mathrm{th}$ of the work in $12$ days. If $A$ can do the work in $k$ days, then find the value of $k$.

$A$ and $B$ can finish a piece of work in $\mathit{}6$ days and $4$ days, respectively, working alone. $A$ started the work. After $2$ days, he joined by $B$. If the time taken to finish the remaining work is $\frac{k}{5}$ days, then find the value of $k$.

Two pipes $A$ and $B$ can separately fill a tank in $20$ minutes and $30$ minutes, respectively, while a third pipe $C$ can empty it in $15$ minutes. If all three pipes are used simultaneously and the tank will be filled in $k$ minutes, then find the value of $k$.

An empty cistern can be filled by two taps $A$ and $B$ in $12$ minutes and $16$ minutes, respectively, and the full cistern can be emptied by a third tap $C$ in $8$ minutes. If all the three taps are turned on simultaneously and in $k$ minutes the empty cistern will be full, then find the value of $k$.

A pipe can fill a tank in $8$ hours. However, in case of outflow from a waste pipe in the bottom, the tank is filled in $12$ hours. If the tank is full and the waste pipe will take $k$ hours to empty it, then find the value of $k$.

A tank can be filled by pipe $A$ in $15$ minutes and by pipe $B$ in $10$ minutes. Another pipe $C$ can empty in $30$ minutes. All three pipes are in operation simultaneously. After $3$ minutes, pipe $C$ is not used. If the remaining tank will be filled by pipes $A$ and $B$ in $\frac{k}{5}$ minutes, then find the value of $k$.