2 men and 5 women can do a piece of work in 4 days, while one man and one woman can finish it in 12 days. How long would it take for 1 man to do the work?

Let's assume that 1 man takes x days to do the work and y days for a women.

So, the amount of work done by 1 man in 1 day = $\dfrac{1}{x}$ and,

the amount of work done by 1 woman in 1 day = $\dfrac{1}{y}$.

∴ The amount of work done by 2 men in 1 day = $\dfrac{2}{x}$ and,

The amount of work done by 5 woman in 1 day = $\dfrac{5}{y}$.

According to the given conditions,

$\dfrac{2}{x} + \dfrac{5}{y} = \dfrac{1}{4}$ …….(i)

$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{12}$ ……(ii)

Multiplying (ii) by 5 we get,

$\dfrac{5}{x} + \dfrac{5}{y} = \dfrac{5}{12}$ …….(iii)

Subtracting (i) from (iii) we get,

$\Rightarrow \dfrac{5}{x} + \dfrac{5}{y} - \Big(\dfrac{2}{x} + \dfrac{5}{y}\Big) = \dfrac{5}{12} - \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{5}{x} - \dfrac{2}{x} = \dfrac{5 - 3}{12} \\[1em] \Rightarrow \dfrac{3}{x} = \dfrac{2}{12} \\[1em] \Rightarrow \dfrac{3}{x} = \dfrac{1}{6} \\[1em] \Rightarrow x = 18.$

Hence, one man can do the work in 18 days.