$2$ men and $5$ boys together can finish a piece of work in $4$ days, while $3$ men and $6$ boys can finish it in $3$ days. Find the time taken: i) by one man alone to finish the work ii) by one boy alone to finish the work.

Let a man's work be $m$ and a boy's work be $b$ per day, leading to the equations $8m + 20b = 1$ and $9m + 18b = 1$. Equating the two yields $8m + 20b = 9m + 18b$, which simplifies to $m = 2b$; substituting this into the first equation gives $16b + 20b = 1$, so $b = 1/36$ and $m = 1/18$. Therefore, one man takes 18 days and one boy takes 36 days to finish the work alone.

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