How many times will the volume of air in the bottle decrease when it is immersed to a depth of 10 meters?

1) Let us write the equation of state for an ideal gas: PV = μRT. Let us express from it the initial gas volume V: V = (μRT) / P. Here P is atmospheric pressure.
2) At a depth of 10 m, the gas pressure will be equal to P + ρgh, where ρgh is the water pressure at a depth of 10 m. Then the volume of gas at a depth of 10 m is: v = (μRT) / (P + ρgh).
3) Find the volume ratio: V / v = ((μRT) / P) / ((μRT) / (P + ρgh)) = (P + ρgh) / P = 1 + ρgh / P.
4) Since P = 100 kPa, ρgh = (1000 * 10 * 10) Pa = 100 kPa, then V / v = 1 + 1/1 = 2.
ANSWER: 2.