Q1: The relationship between void ratio (e) and porosity ratio (n) is :

ANS:D - e = n/(1 - n)

e = vv/vs.
n = vv/vt.

Where vv = volume of void,
vs = volume of solid,
vt = total volume.

But from relation vt=vv+vs.
So vs=vt-vv and vv=n*vt.

Then substitute value of vv,
e = n*vt/vs.

Again substitute value of vs enterms vs=vt-vv.

e = n*vt/(vt-vv) and also divided by vt.
e = n*(vt/vt)/((vt/vt)-(vv/vt)).
e = n/(1-vv/vt) but vv/vt = n.

There for e = n/(1-n). First, e=Vv/Vs -------------> (1)

V=Vv+Vs; from here, Vs=V-Vv.


From (1) above, e=Vv/V-Vv; now dividing numerator and denominator by V knowing that Vv/V=n; where n=porosity of the soil.
Hence, e=(Vv/V)/(V/V-Vv/V),
e=n/(1-n). e = Vv/Vs.

But V = Vv+Vs, Vs = V-Vv.
e = Vv/(V-Vv).

Divide throughout by V.

e = (Vv/v)/((V-Vv)/V) as Vv/V = n.
e = n/(1-n)