Question

$2$ women and $5$ men can completed a piece of work in $4$ days if the same work is completed by $3$ women and six men will take $3$ days complete find the time taken by one man and one women to do the same work?

Answer 1

Let one women takes $x$ days to finish the work and one man takes $y$ days to finish the work .

According to the question.

$\begin{array}{c}\frac{2}{x}+\frac{5}{y}=\frac{1}{4}\\ 4\left(2y+5x\right)=xy\\ 8y+20x=xy......\left(1\right)\end{array}$

Now,

$\begin{array}{c}\frac{3}{x}+\frac{6}{y}=\frac{1}{3}\\ \Rightarrow 9y+18x=xy.........\left(2\right)\\ \left(1\right)\times 9-\left(2\right)\times 8\\ \left(8y+20x=xy\right)\mathrm{9...............}\left(3\right)\\ \left(9y+18x=xy\right)\mathrm{8...............}\left(4\right)\\ eq\left(3\right)-\left(4\right)\\ 72y+18x=9xy\\ 72x+144x=8xy\\ \underset{–}{---}\\ 36x=xy\\ y=36\end{array}$

Putting $y=36$ in equation $\left(1\right)$

We get $x=18$

One man and one women will do work in $d$ days.

$\frac{1}{36}+\frac{1}{18}=\frac{1}{d}$

Hence,

$d=12$ days.