A gas is filled in a vessel at a certain temperature and pressure. At the same temperature more gas is filled in the vessel so that its mass increased by 40%. Calculate the ratio of initial and final pressures.

Explanation

To solve the problem, we need to find the ratio of initial pressure (P1) to final pressure (P2) of a gas when its mass is increased by 40% while keeping the temperature constant.

### Step-by-Step Solution:

1. **Understand the relationship between pressure and mass**:
According to the ideal gas law, pressure (P) is directly proportional to the mass (m) of the gas when the volume (V) and temperature (T) are constant. This can be expressed as:
\[ P \propto m \]

2. **Define the initial and final mass**:
Let the initial mass of the gas be \( m_i \). The final mass after increasing by 40% can be expressed as:
\[ m_f = m_i + 0.4 m_i = 1.4 m_i \]

3. **Express the initial and final pressures**:
Using the relationship from step 1, we can write:
\[ P_1 \propto m_i \quad \text{and} \quad P_2 \propto m_f = 1.4 m_i \]

4. **Set up the ratio of pressures**:
The ratio of initial pressure to final pressure can be expressed as:
\[ \frac{P_1}{P_2} = \frac{m_i}{m_f} = \frac{m_i}{1.4 m_i} \]

5. **Simplify the ratio**:
The mass terms \( m_i \) cancel out:
\[ \frac{P_1}{P_2} = \frac{1}{1.4} = \frac{10}{14} = \frac{5}{7} \]

6. **Conclusion**:
Therefore, the ratio of initial pressure to final pressure is:
\[ \frac{P_1}{P_2} = \frac{5}{7} \]

### Final Answer:
The ratio of initial and final pressures is \( \frac{5}{7} \).