When equal volumes of three liquids of densities `p_1, p_2, p_3` are mixed then the resultant density of the mixture will be:

Options

A. `(p_(1)+p_(2)+p_(3))/(3)`
B. `(p_(1)+p_(2)+p_(3))/(6)`
C. `(p_(1)^2+p_(2)^2+p_(3)^2)/(6)`
D. `p_(1)+p_(2)+p_(3)`

Answer

To find the resultant density of a mixture of three liquids with equal volumes and densities \( p_1, p_2, \) and \( p_3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Density**:
Density (\( \rho \)) is defined as mass (\( m \)) per unit volume (\( V \)):
\[ \rho = \frac{m}{V} \]
Therefore, mass can be expressed as:
\[ m = \rho \cdot V \]

2. **Assume Equal Volumes**:
Let the volume of each liquid be \( V \). Since we have three liquids, the total volume of the mixture will be:
\[ V_{\text{total}} = V + V + V = 3V \]

3. **Calculate the Mass of Each Liquid**:
Using the formula for mass:
- Mass of liquid 1:
\[ m_1 = p_1 \cdot V \]
- Mass of liquid 2:
\[ m_2 = p_2 \cdot V \]
- Mass of liquid 3:
\[ m_3 = p_3 \cdot V \]

4. **Calculate the Total Mass**:
The total mass of the mixture is the sum of the masses of the three liquids:
\[ m_{\text{total}} = m_1 + m_2 + m_3 = p_1 \cdot V + p_2 \cdot V + p_3 \cdot V \]
Factoring out \( V \):
\[ m_{\text{total}} = (p_1 + p_2 + p_3) \cdot V \]

5. **Calculate the Resultant Density**:
The resultant density (\( \rho_{\text{resultant}} \)) of the mixture is given by the total mass divided by the total volume:
\[ \rho_{\text{resultant}} = \frac{m_{\text{total}}}{V_{\text{total}}} = \frac{(p_1 + p_2 + p_3) \cdot V}{3V} \]
Simplifying this expression:
\[ \rho_{\text{resultant}} = \frac{p_1 + p_2 + p_3}{3} \]

### Final Result:
The resultant density of the mixture of the three liquids is:
\[ \rho_{\text{resultant}} = \frac{p_1 + p_2 + p_3}{3} \]