$P$ is twice as good a workman as $Q,$ and together they finish a work in $16$ days. In how many days can it be done by each separately?

There is sufficient food for $400$ men for $31$ days. After $28$ days,$280$ men leave the place. For how many days will the rest of the food last for the rest of the men? (in days)

$A$ and $B$ can do a work in $45 \text{ days}$ and $40 \text{ days}$,respectively. They began the work together,but $A$ left after some time and $B$ completed the remaining work in $23 \text{ days}$. After how many days of the start of the work did $A$ leave?

$A$ track of $100\, m$ can be built by $7\, men$ or $10\, women$ in $70\, days.$ How many days will $14\, men$ and $20\, women$ take to build a track of $600\, m$?

$A$ man is twice as fast as a woman and a woman is twice as fast as a boy in doing a work. If all of them, a man, a woman and a boy, can finish the work in $7$ days, in how many days will a boy do it alone?

$18$ boys can do a piece of work in $24$ days. In how many days can $27$ boys do the same work?