$($b$)$ For a given soil sample show that

i$)Y_{\text{s}\text{a}\text{t}}=Y_d+$ n$.Y_w$

ii$)Y_d=e.S.\frac{Y_w}{1+e}w$

iii$)Y_{\text{s}\text{a}\text{t}}=[(1-n)G_s+n].Y_w$

where, $Y_{\text{s}\text{a}\text{t}}=$ Saturated Unit weight, $Y_d=$ Dry unit weight, $n=$ Porosity, $e=$ Void Ratio

$S=$ Degree of saturation, $Y_w=$ Unit weight of water, $G_s=$ Specific Gravity of solid