Three samples A, B and C of the same gas `(gamma = 1.5)` have equal volumes and temperatures. The volume of each sample is doubled, the process being isothermal for A, adiabtic for B and isobaric for C. If the final pressures are equal for the three samples, Find the ratio of the initial pressures.

Explanation

To solve the problem, we need to analyze the three processes (isothermal, adiabatic, and isobaric) and use the ideal gas law and the properties of each process to find the initial pressures of the gas samples A, B, and C.

### Step 1: Understand the initial conditions
Let the initial pressures of samples A, B, and C be \( P_A \), \( P_B \), and \( P_C \) respectively. The initial volume \( V \) and temperature \( T \) are the same for all three samples.

### Step 2: Analyze the isothermal process for sample A
In an isothermal process, the pressure and volume are related by the equation:
\[
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