Pooja and Ritu can do a piece of work in $17\dfrac{1}{7}$ days. If one day work of Pooja be three fourth of one day work of Ritu; find in how many days each will do the work alone.

Let Pooja alone can do work in x days and Ritu alone can do in y days.

So, in one day Pooja can do $\dfrac{1}{x}$ th part of work and Ritu can do $\dfrac{1}{y}$ th part of work.

Given,

One day work of Pooja equals three fourth of one day work of Ritu.

$\therefore \dfrac{1}{x} = \dfrac{3}{4} \times \dfrac{1}{y} \\[1em] \Rightarrow \dfrac{1}{x} = \dfrac{3}{4y} \\[1em] \Rightarrow x = \dfrac{4y}{3} …….(1)$

Given,

Pooja and Ritu can do a piece of work in $17\dfrac{1}{7} \text{ or } \dfrac{120}{7}$ days.

So, in one day both of them can do $\dfrac{7}{120}$ th part of work.

$\therefore \dfrac{1}{x} + \dfrac{1}{y} = \dfrac{7}{120}$

Substituting value of x from equation (1) in above equation :

$\Rightarrow \dfrac{1}{\dfrac{4y}{3}} + \dfrac{1}{y} = \dfrac{7}{120} \\[1em] \Rightarrow \dfrac{3}{4y} + \dfrac{1}{y} = \dfrac{7}{120} \\[1em] \Rightarrow \dfrac{3 + 4}{4y} = \dfrac{7}{120} \\[1em] \Rightarrow \dfrac{7}{4y} = \dfrac{7}{120} \\[1em] \Rightarrow y = \dfrac{120 \times 7}{7 \times 4} \\[1em] \Rightarrow y = 30.$

Substituting value of y in equation (1), we get :

$\Rightarrow x = \dfrac{4 \times 30}{3} \\[1em] = 40.$

Hence, Pooja can do the work alone in 40 days and Ritu in 30 days.