Two liquids `A` and `B` of equal volumes have their specific heats in the ration`2 : 3`. If they have same thermal capacity, then the ratio of their densities is

Options

A. `1 : 1`
B. `2 : 3`
C. `3 : 2`
D. `5 : 6`

Explanation

To solve the problem, we need to find the ratio of the densities of two liquids A and B given that they have equal volumes, specific heats in the ratio of 2:3, and the same thermal capacity.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Let the specific heat of liquid A be \( s_A \) and that of liquid B be \( s_B \).
- According to the problem, the ratio of their specific heats is given as:
\[ \frac{s_A}{s_B} = \frac{2}{3} \]

2. **Define Thermal Capacity:**
- The thermal capacity \( C \) of a substance is defined as:
\[ C = m \cdot s \]
- Where \( m \) is the mass and \( s \) is the specific heat.

3. **Set Up the Equation for Thermal Capacities:**
- Since the thermal capacities of both liquids are the same, we can write:
\[ C_A = C_B \]
- This means:
\[ m_A \cdot s_A = m_B \cdot s_B \]

4. **Express Mass in Terms of Density and Volume:**
- The mass of each liquid can be expressed as:
\[ m_A = \rho_A \cdot V_A \quad \text{and} \quad m_B = \rho_B \cdot V_B \]
- Since the volumes are equal (\( V_A = V_B \)), we can denote the volume as \( V \):
\[ m_A = \rho_A \cdot V \quad \text{and} \quad m_B = \rho_B \cdot V \]

5. **Substitute Mass into the Thermal Capacity Equation:**
- Substituting the expressions for mass into the thermal capacity equation gives:
\[ \rho_A \cdot V \cdot s_A = \rho_B \cdot V \cdot s_B \]
- Since \( V \) is common and non-zero, we can cancel it out:
\[ \rho_A \cdot s_A = \rho_B \cdot s_B \]

6. **Rearranging the Equation:**
- Rearranging the equation to find the ratio of densities:
\[ \frac{\rho_A}{\rho_B} = \frac{s_B}{s_A} \]

7. **Substituting the Ratio of Specific Heats:**
- From the ratio of specific heats \( \frac{s_A}{s_B} = \frac{2}{3} \), we can find \( \frac{s_B}{s_A} \):
\[ \frac{s_B}{s_A} = \frac{3}{2} \]

8. **Final Calculation of Density Ratio:**
- Therefore, substituting this back into the density ratio gives:
\[ \frac{\rho_A}{\rho_B} = \frac{3}{2} \]

### Conclusion:
The ratio of the densities of liquids A and B is:
\[ \rho_A : \rho_B = 3 : 2 \]