The density index I_D in terms of densities is given by _____
a) \(\frac{(γ_{d,max}-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d}) γ_d}\)
b) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d,min}) γ_d}\)
c) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_d-γ_{d,max}) γ_d}\)
d) \(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d}) γ_d}\)

Answer: b
Explanation: The density index is given by,
I_D=\(\frac{(e_{max} – e)}{(e_{max}-e_{min})}\)
\(e=(\frac{Gγ_w}{γ_d})-1\)
\(e_{max}=(\frac{Gγ_w}{γ_{d,min}})-1\)
\(e_{min}=(\frac{Gγ_w}{γ_{d,max}})-1\)
substituting the values of voids ratio
I_D=\(\frac{((\frac{Gγ_w}{γ_{d,min}}) – (\frac{Gγ_w}{γ_d}))}{((\frac{Gγ_w}{γ_{d,min}})- (\frac{Gγ_w}{γ_{d,max}}))}\)
I_D=\(\frac{(γ_d-γ_{d,min}) γ_{d,max}}{(γ_{d,max}-γ_{d,min}) γ_d}.\)