Two substances of densities ρ1 and ρ2 are mixed in equal volume and their relative density is 4. When they are mixed in equal masses, relative density is 3. The values of ρ1 and ρ2 respectively are
- (a)6, 2
- (b)3, 5
- (c)12, 4
- (d)9, 3
Correct - A, 6 and 2. Mixing equal VOLUMES makes the mean density the simple arithmetic average: (rho1 + rho2)/2 = 4, so rho1 + rho2 = 8. Mixing equal MASSES makes the mean density the harmonic mean: 2*rho1*rho2/(rho1 + rho2) = 3, and with rho1 + rho2 = 8 this gives rho1*rho2 = 12. The two numbers that add to 8 and multiply to 12 are 6 and 2, so (rho1, rho2) = (6, 2).
- (b)3, 5 — Their sum is 8 (fitting the equal-volume condition) but their product is 15, giving an equal-mass relative density of 2*15/8 = 3.75, not 3 - so it fails the second condition.
- (c)12, 4 — Their sum is 16, so the equal-volume average is 8, not 4 - it fails the first condition immediately.
- (d)9, 3 — Their sum is 12, so the equal-volume average is 6, not 4 - again failing the first condition.
When two substances are mixed by equal volume, the resulting density is the arithmetic mean of the two densities; when mixed by equal mass, it is the harmonic mean. Writing down these two averages gives two equations that pin down both densities.
The key insight is that equal-volume mixing averages the densities, while equal-mass mixing averages the specific volumes (a harmonic mean of the densities). Turn the two relative-density statements into a sum and a product, then solve the quadratic.
- Mixing equal volumes gives a mean density equal to (rho1 + rho2)/2, the arithmetic mean.
- Mixing equal masses gives a mean density equal to 2*rho1*rho2/(rho1 + rho2), the harmonic mean.
- Relative density (specific gravity) is a substance's density divided by that of water, a pure number.
- Here rho1 + rho2 = 8 and rho1*rho2 = 12, whose solution is the pair 6 and 2.
- Using the arithmetic mean for the equal-mass case (it needs the harmonic mean).
- Forgetting that relative density is dimensionless, so the same arithmetic applies as for density.
Asked by giving mixture densities for equal-volume and equal-mass mixing and asking for the component densities.
No directly related past PYQ was found.
- practice — not a real PYQ
Two liquids of relative densities 8 and 2 are mixed in equal volumes. The relative density of the mixture is
- (a)4
- (b)5
- (c)3.2
- (d)6
Answer(b) 5 - the arithmetic mean (8 + 2)/2 = 5.
- practice — not a real PYQ
Two liquids of relative densities 6 and 2 are mixed in equal masses. The relative density of the mixture is
- (a)4
- (b)3
- (c)2.5
- (d)8
Answer(b) 3 - the harmonic mean 2*6*2/(6 + 2) = 3.