Liquid of volume V₁ and density ρ₁ is mixed with another liquid of volume V₂ and density ρ₂

Problem

A liquid of volume V₁ and density ρ₁ is mixed with another liquid of volume V₂ 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 

In this scenario, you are given two liquids with volumes

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

Let's denote the total volume before mixing as ${�}_{\text{total}}$: ${�}_{\text{total}}={�}_1+{�}_2$

Now, the density ($�$) of a substance is defined as its mass ($�$) divided by its volume ($�$): $�=\frac{�}{�}$

We know that the mixture volume (${�}_{\text{mix}}$) is equal to the total volume before mixing, so ${�}_{\text{mix}}={�}_{\text{total}}$.

The total mass (${�}_{\text{total}}$) of the mixture is the sum of the masses of the individual liquids: ${�}_{\text{total}}={�}_1\cdot {�}_1+{�}_2\cdot {�}_2$

Now, substitute these values into the expression for density ($�$): ${�}_{\text{mix}}=\frac{{�}_{\text{total}}}{{�}_{\text{mix}}}$

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

So, the density of the mixture is given by this expression.

Given parameters:

• Volume of the first liquid: ${�}_1$
• Density of the first liquid: ${�}_1$
• Volume of the second liquid: ${�}_2$
• Density of the second liquid: ${�}_2$

Total volume before mixing: ${�}_{\text{total}}={�}_1+{�}_2$

The density of the mixture: ${�}_{\text{mix}}=\frac{{�}_1\cdot {�}_1+{�}_2\cdot {�}_2}{{�}_1+{�}_2}$