A and B together can do a piece of work in 15 days. If A's one day work is $1\dfrac{1}{2}$ times the one day's work of B, find in how many days can each do the work.

Let A's one day work be x and B's one day work be y.

According to first condition given in the problem,

x = $\dfrac{3}{2}$y

2x = 3y

2x - 3y = 0 …….(i)

Also given, A and B together can do a piece of work in 15 days.

∴ x + y = $\dfrac{1}{15}$

⇒ 15(x + y) = 1

⇒ 15x + 15y = 1 …….(ii)

Multiplying (i) by 5 we get,

10x - 15y = 0 ……(iii)

Adding (ii) and (iii) we get,

⇒ 15x + 15y + 10x - 15y = 1 + 0

⇒ 25x = 1

⇒ x = $\dfrac{1}{25}$.

Substituting x in (i) we get,

$\Rightarrow 2 \times \dfrac{1}{25} - 3y = 0 \\[1em] \Rightarrow 3y = \dfrac{2}{25} \\[1em] \Rightarrow y = \dfrac{2}{75}.$

Since, A's one day work is x and B's one day work is y, so A can do complete work in $\dfrac{1}{x}$ and B can do work in $\dfrac{1}{y}$ days.

$\dfrac{1}{x} = \dfrac{1}{\dfrac{1}{25}} = 25$ days.

$\dfrac{1}{y} = \dfrac{1}{\dfrac{2}{75}} = \dfrac{75}{2} = 37\dfrac{1}{2}$ days.

Hence, A will do the work in 25 days and B will do the work in $37\dfrac{1}{2}$ days.