$X,Y,W,Z$ jointly purchased an office space for Rs $84$ lakhs. The amount contributed by $Y,W$ and $Z$ put together is three times of $X$. The amount contributed by $X,W$ and $Z$ put together is $320%$ of $Y$. Further, the amount contributed by $W$ is $20%$ of $X,Y$ and $Z$ put together. What is the contribution of $Z$?

Let the contributions of $X,Y,W,Z$ be $x,y,w,z$ lakhs respectively.

Given,

$x+y+w+z=84$ .....$(1)$

Also,

$y+w+z=3x$ .....$(2)$

$x+w+z=320%$ of $y$

$x+w+z=\frac{320}{100}y$

$x+w+z=\frac{16}{5}y$ .....$(3)$

Also,

$w=20%$ of $(x+y+z)$

$w=\frac{1}{5}(x+y+z)$ .....$(4)$

From equation $(1)$ and $(2)$:

$84-x=3x$

$4x=84$

$x=21$

From equation $(1)$ and $(3)$:

$84-y=\frac{16}{5}y$

$420-5y=16y$

$21y=420$

$y=20$

From equation $(4)$:

$w=\frac{1}{5}(21+20+z)$

$5w=41+z$ .....$(5)$

Now using total:

$21+20+w+z=84$

$w+z=43$ .....$(6)$

From $(6)$,

$z=43-w$

Put in $(5)$:

$5w=41+43-w$

$5w=84-w$

$6w=84$

$w=14$

Now,

$z=43-14$

$z=29$

Therefore, the contribution of $Z$ is Rs $29$ lakhs.