Given,
Porosity n=30%=0.3
Specific gravity G=2.7
Voids ratio e = n / (1-n)
e=0.3/(1-0.3)=0.429
Dry unit weight ?_d = G?_w/(1-e)
?_d=(2.7*9.81)/(11+0.429)
?_d=18.535 kN/m³.

In a fully saturated sample, volume of voids gets completely filled by water leaving no air voids present in sample. Hence, volume of voids equals volume of water.

The relative compaction is given by,
\(R_c = \frac{?_d}{?_{d,max}}\)
Since ?_d=?_s(1+e_min)
?_d,max=?_s(1+e)
? \(R_c= \frac{?_d}{?_{d,max}}= \frac{?_s(1+e_{min})}{?_s(1+e)} \)
\(R_c= \frac{(1+e_{min})}{(1+e)}. \)

The submerged unit weight is given by,
?? = ?_sat ? ?_w
?? = \(\frac{(G+e) ?_w}{(1+e)} ? ?_w\)
? \(??=\frac{(G-1) ?_w}{(1+e)}\).

Solids and water, solids and air are two-phase systems. Sand, gravel and air are not phases. Solids, water and air are considered to be a three-phase system of soil mass.

The soil will have maximum possible voids when soil grains are arranged in a cubical array of spheres. In a cubical array, angle of orientation ? = 90°.

Given,
Void ratio e=0.35
Porosity \(n = \frac{e}{1+e}\)
\(n = \frac{0.35}{1+0.35}\)
\(n = \frac{0.35}{1.35}\)
\(\)n=0.259
In percentage n=25.9%.

Given,
Dry density ?_d = 18.5 kN/m³
Specific gravity G=2.7
Void ratio e= 0.6
Degree of saturation = 50% = 0.5
Since, e = (W*G)/S
W = (e*S)/G=(0.6*0.5)/2.7=0.111
Bulk unit weight ?=?_d(1+W)
?=18.5*1.111=20.554 kN/m³.

Concept:

Void ratio (e): This is defined as ratio of volume of voids (V_v) in sample to  volume of soil solids (V_s). 

$e=\frac{V_v}{V_s}$

Porosity (n): This is defined as ratio of volume of voids (V_v) in sample to total volume of soil (V), expressed in percentage.

$n=\frac{V_v}{V}$

$n\mathrm{\%}=\frac{V_v}{V}×100$

Calculation:

Given

Total volume of soil sample (V) = 50 cm³

The soil sample displaces 30 cm³ of water which means volume of soil solids is 30 cm³.

Volume of soil solids (V_s) = 30 cm³

Volume of voids (V_v) = 50 - 30 = 20 cm³

Porosity (n) = $\frac{V_v}{V}=\frac{20}{50}=0.4$

$Voidratio(e)=\frac{V_v}{V_s}=\frac{20}{30}=0.666$

Void ratio (e) = 0.666 ≈ 0.67

Hence option (3) is correct.

Given,
G=2.7
e=0.66
w=20%=0.2
Since degree of saturation is S=wG/e
S=(0.2*2.7)/0.66
S=0.8181
In percentage S=81.81%.