The specific gravity of a mixture of two substances of equal volumes is 4. The specific gravity of the mixture becomes 3 when these two substances are mixed in equal masses. Calculate the specific gravity of the substances.

Explanation

Let the specific gravities of the two substances be `S_(1)andS_(2)` and the density of water at `4^(@)C` be `rho`.
`therefore` Densities of the substances are `S_(1)rhoandS_(2)rho`.
Let V = volume of each of the substances in the mixture.
`therefore` Total mass of the mixture = `(VS_(1)rho+VS_(2)rho)`
Total volume of the mixture = V + V = 2V
`therefore` Density of the mixture = `(VS_(1)rho+VS_(2)rho)/(2V)=(rho(S_(1)+S_(2)))/2`
`therefore` Specific gravity of the mixture
`=(S_(1)+S_(2))/2=4` (according to the problem)
or, `S_(1)+S_(2)=8" "...(1)`
Again, let m = mass of each of the two substances in the mixture.
`therefore` Total mass of the mixture = m + m = 2m
Total volume of the mixture = `(m/(S_(1)rho)+m/(S_(2)rho))`
`therefore` Density of the mixture = `(2m)/(m/(S_(1)rho)+m/(S_(2)rho))=(2S_(1)S_(2)rho)/(S_(1)+S_(2))`
`therefore` Specific gravity of the mixture
`=(2S_(1)S_(2))/(S_(1)+S_(2))=3` (according to the problem)
or, `S_(1)S_(2)=3/2(S_(1)+S_(2))=3/2xx8=12`
`therefore" "S_(1)-S_(2)=sqrt((S_(1)+S_(2))^(2)-4S_(1)S_(2))`
`=sqrt((8)^(2)-4xx12)=4" "...(2)`
From equations (1) and (2), `S_(1)=6andS_(2)=2`
`therefore` The specific gravities of the two given substances are 6 and 2.