Step 1: Recall the proportionality form of Stokes' law:
Rather than writing out the full Stokes' law expression, it is enough to remember that for particles of the same material settling in the same fluid under Stokes' (laminar, creeping flow) conditions, the settling velocity depends on particle size only through the square of the diameter, that is \[ v \propto d^{2} \] All the other factors, particle density, fluid density, fluid viscosity and gravitational acceleration, cancel out when comparing two batches of the same barite in the same mud.
Step 2: Express both settling velocities using a common proportionality constant:
Let \(v = k d^{2}\) for some constant \(k\) that is identical for both batches since the fluid and solid properties do not change. Then for batch P, \(v_{P} = k (75)^{2}\), and for batch Q, \(v_{Q} = k (25)^{2}\), both diameters measured in the same micrometer units so the constant \(k\) cancels cleanly in the ratio.
Step 3: Divide to find how many times faster batch P settles:
Dividing, \[ \frac{v_{P}}{v_{Q}} = \frac{(75)^{2}}{(25)^{2}} = \frac{5625}{625} = 9 \] which confirms, by direct numerical computation rather than the shortcut ratio method, that batch P settles nine times faster than batch Q.
Step 4: Sanity check the result:
Batch P particles are three times larger in diameter than batch Q particles (75 divided by 25 equals 3), and since settling velocity scales with the square of size, tripling the diameter should scale the velocity by three squared, which is nine. This agrees with the calculation and confirms option (B) is correct.
Final Answer:
\[ \boxed{\dfrac{v_{P}}{v_{Q}} = 9\ \text{(Option B, Nine)}} \]