If two masses A and B have their masses in the ratio 1:4 and their volumes are equal, then the densities have the ratio

Options

A. `1:4`
B. `4:1`
C. `2:1`
D. `3:1`

Answer

To solve the problem, we need to find the ratio of the densities of two masses A and B, given that their masses are in the ratio of 1:4 and their volumes are equal.

### Step-by-Step Solution:

1. **Define the Masses**:
Let the mass of body A be \( m_A \) and the mass of body B be \( m_B \). According to the problem, the ratio of their masses is:
\[ \frac{m_A}{m_B} = \frac{1}{4} \]
This implies that \( m_A = 1k \) and \( m_B = 4k \) for some constant \( k \).

2. **Define the Volumes**:
Since the volumes of both bodies are equal, let the volume of each body be \( V \).

3. **Use the Density Formula**:
The density (\( \rho \)) of an object is given by the formula:
\[ \rho = \frac{m}{V} \]
Therefore, the densities of bodies A and B can be expressed as:
\[ \rho_A = \frac{m_A}{V} \quad \text{and} \quad \rho_B = \frac{m_B}{V} \]

4. **Substitute the Masses**:
Substitute the expressions for \( m_A \) and \( m_B \) into the density formulas:
\[ \rho_A = \frac{1k}{V} \quad \text{and} \quad \rho_B = \frac{4k}{V} \]

5. **Find the Ratio of Densities**:
Now, we can find the ratio of the densities:
\[ \frac{\rho_A}{\rho_B} = \frac{\frac{1k}{V}}{\frac{4k}{V}} = \frac{1k}{4k} = \frac{1}{4} \]

6. **Conclusion**:
Thus, the ratio of the densities of masses A and B is:
\[ \frac{\rho_A}{\rho_B} = \frac{1}{4} \]

### Final Answer:
The ratio of the densities of masses A and B is \( 1:4 \).

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