4 men and 10 women were put on a work. They completed $$\frac{1}{3}$$ of the work in 4 days. After this 2 men and 2 women were increased. They completed $$\frac{2}{9}$$ more of the work in 2 days. If the remaining work is to be completed in 3 days, then how many more women must be increased ?

Solution(By Examveda Team)

$$\eqalign{ & {\text{Let 1 man's 1 day's work}} = x \cr & {\text{And }} \cr & {\text{1 women's 1 day's work}} = y \cr & {\text{Then,}} \cr & \Rightarrow 4x + 10y = \frac{1}{3} \times \frac{1}{4} = \frac{1}{{12}} \cr & \Rightarrow 4x + 10y = \frac{1}{{12}} \cr & \Rightarrow 2x + 5y = \frac{1}{{24}}.....(i) \cr & {\text{And,}} \cr & \Rightarrow 6x + 12y = \frac{1}{9} \cr & \Rightarrow 2x + 4y = \frac{1}{{27}}.....({\text{ii}}) \cr} $$
Subtracting (ii) from (i), we get
$$\eqalign{ & y = \frac{1}{{24}} - \frac{1}{{27}} = \frac{1}{{216}} \cr & {\text{Now,}} \cr} $$
Now,
(6 men + 12 women)'s 3 day's work
$$\eqalign{ & = \left( {\frac{1}{9} \times 3} \right) \cr & = \frac{1}{3} \cr & {\text{Work completed}} \cr & = \left( {\frac{1}{3} + \frac{2}{9} + \frac{1}{3}} \right) \cr & = \frac{8}{9} \cr & \therefore {\text{Remainig work}} \cr & = \left( {1 - \frac{8}{9}} \right) \cr & = \frac{1}{9} \cr & {\text{1 women's 3 day's work}} \cr & = \left( {\frac{1}{{216}} \times 3} \right) \cr & = \frac{1}{{72}} \cr} $$
In 3 day's $$\frac{1}{{72}}$$ work is done by 1 women.
$$\eqalign{ & \therefore {\text{In 3 day's }}\frac{1}{9}{\text{work is done by}} \cr & {\text{ = }}\left( {72 \times \frac{1}{9}} \right) \cr & = {\text{8 women}}{\text{.}} \cr} $$