What is the density of the mixture?

Problem
A liquid of mass m₁ and density ρ₁ is mixed with another liquid of mass m₂ and density ρ₂. After mixing liquids, the mixture volume is equal to the total volume of liquids before mixing. What is the density of the mixture? Solution

When two liquids are mixed, the density of the resulting mixture depends on the masses and densities of the individual liquids. In this case, you are given two liquids with masses 

${�}_1$ and ${�}_2$ and densities ${�}_1$ and ${�}_2$, and the mixture volume is equal to the total volume of the liquids before mixing.

The density ($�$) of a substance is defined as its mass ($�$) divided by its volume ($�$):

  $�=\frac{�}{�}$

The total volume (${�}_{\text{total}}$) before mixing is the sum of the individual volumes (${�}_1$ and ${�}_2$) of the liquids:

  ${�}_{\text{total}}={�}_1+{�}_2$

Since the mixture volume is equal to the total volume, we can write:

  ${�}_{\text{mix}}={�}_{\text{total}}$

Now, let's express the volumes in terms of masses and densities:

  ${�}_1=\frac{{�}_1}{{�}_1}$,

 ${�}_2=\frac{{�}_2}{{�}_2}$

Substitute these expressions back into the total volume equation:

  ${�}_{\text{total}}=\frac{{�}_1}{{�}_1}+\frac{{�}_2}{{�}_2}$

Now, we know that ${�}_{\text{mix}}={�}_{\text{total}}$, so:

  ${�}_{\text{mix}}=\frac{{�}_1}{{�}_1}+\frac{{�}_2}{{�}_2}$

The density of the mixture (${�}_{\text{mix}}$) is given by the total mass of the mixture (${�}_{\text{total}}$) divided by the mixture volume (${�}_{\text{mix}}$):

  ${�}_{\text{mix}}=\frac{{�}_{\text{total}}}{{�}_{\text{mix}}}$

Since the total mass (${�}_{\text{total}}$) is the sum of the masses of the individual liquids, we have:

  ${�}_{\text{total}}={�}_1+{�}_2$

Now, substitute this into the expression for ${�}_{\text{mix}}$:

  ${�}_{\text{mix}}=\frac{{�}_1+{�}_2}{\frac{{�}_1}{{�}_1}+\frac{{�}_2}{{�}_2}}$