(i) Prove that the density of the mixture of two substances with densities `rho_(1)andrho_(2)` of equal mass will be `(2rho_(1)rho_(2))/(rho_(1)+rho_(2))`.
(ii) Prove that if the two substance with densities `rho_(1)andrho_(2)` are mixed in equal volumes, then the density of the mixture thus formed will be `1/2(rho_(1)+rho_(2))`

Explanation

(i) Let the mass of each of the substances be m.
`therefore` Mass of the mixture = m + m = 2m
Volume of the mixture = `(m/rho_(1)+m/rho_(2))`
`therefore` Density of the mixture = `(2m)/(m/rho_(1)+m/rho_(2))=(2rho_(1)rho_(2))/(rho_(1)+rho_(2))`
(ii) Let the volume of each of the substances be V.
`therefore` Total mass of the mixture = `(Vrho_(1)+Vrho_(2))`
Total volume of the mixture = (V + V) = 2V
`therefore` Density of the mixture = `(Vrho_(1)+Vrho_(2))/(2V)=1/2(rho_(1)+rho_(2))`