What is the effective density of the maxture of water and milk when
(i) they are taken with same masses
(ii) they are taken with same volumes

Explanation

Let us say the densities of water and milk are `rho_(1) and rho _(2)`
(i) When they are taken with same mass .m. and their volumes are `V_(1) and V_(2),` the mass of water `m = rho _(1) V_(1) , V_(1) =(m)/(rho_(1))` and the mass of milk `m = rho_(2) V_(2), V_(2) = (m)/( rho _(2))`
Total mass of water and milk is `m + m 2m`
Total volume of the water and milk is `V _(1) + V _(2) = (m)/( rho _(1)) + (m)/( rho _(2))`
`= m ((1)/( rho_(1)) + (1)/( rho _(2 )))`
` = (m ( rho _(1) + rho _(2)))/(rho _(1) rho _(2))`
The effective density of the mixture `(rho _(eff))=` Total mass /Total volume
`= (2m)/( m (rho _(1) + rho _(2)) // rho _(1) rho _(2))`
`= ( 2)/(( rho _(1)+ rho _(2)) // rho_(1) rho _(2))`
` = (2 rho _(1) rho_(2))/( rho_(1) + rho _(2))`
(ii) when they are taken with same volume .V. and their masses are `m _(1) and m _(2)` the volume of water `V = m _(1) // rho _(1)`
That is `m _(1) = V rho _(1)`
and the volume of milk `V = m _(2) // rho _(2)`
Total mass of water and milk is `m _(1) + m _(2) = V rho _(1) + V rho _(2)`
`=V (rho _(1) + rho _(2))`
Total volume of water and milk is `V +V =2V`
The effective density of the mixture `(rho _(eff))=`Total mass/Total volume
`rho _(eff) =(V (rho _(1) + rho _(2))/( 2V)`
`= 1/2 (rho _(1) + rho _(2))`