$A$ alone would take $27 \, \text{days}$ more to complete the job than if both $A$ and $B$ worked together. If $B$ worked alone, he would take $3 \, \text{days}$ more to complete the job than $A$ and $B$ working together. What time would they take if both $A$ and $B$ worked together? (in $\text{days}$)

$A$ $10\, \text{hectare}$ field is reaped by $2\, \text{men},$ $3\, \text{women},$ and $4\, \text{children}$ together in $10\, \text{days}.$ If the working capabilities of a man, a woman, and a child are in the ratio $5: 4: 2,$ then a $16\, \text{hectare}$ field will be reaped by $6\, \text{men},$ $4\, \text{women},$ and $7\, \text{children}$ in how many days?

$A$ contract is to be completed in $92$ days and $234$ men were set to work,each working $16$ hours a day. After $66$ days,$\frac{4}{7}$ of the work is completed. How many additional men may be employed,so that the work may be completed in time,each man now working $18$ hours a day?

$P$ and $Q$ can do a piece of work in $30$ and $40$ days respectively. They began the work together,but $P$ left after some days and $Q$ finished the remaining work in $12$ days. After how many days did $P$ leave?

$A$ and $B$ can complete a piece of work in $18$ days; $B$ and $C$ in $24$ days; $C$ and $A$ in $36$ days. $A$ alone can complete the work in (in days):

$A$ and $B$ can do a piece of work in $8$ days,$B$ and $C$ can do the same work in $12$ days,and $A, B,$ and $C$ together complete it in $6$ days. The number of days required to finish the work by $A$ and $C$ together is: