Two trains, A and B, are travelling towards each other on parallel tracks. Train A is 150 m long and travelling at 164 km/h. Train B is 100 m long and travelling at 88 km/h. If the front of Train A is

Train A speed: $164 \text{ km/h}$
Train B speed: $88 \text{ km/h}$

$$1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}$$

Speed of Train A ($V_A$):

$$V_A = 164 \times \frac{5}{18} = \frac{820}{18} = \frac{410}{9} \text{ m/s}$$

Speed of Train B ($V_B$):

$$V_B = 88 \times \frac{5}{18} = \frac{440}{18} = \frac{220}{9} \text{ m/s}$$

The trains are initially $240 \text{ m}$ apart.
Length of Train A ($L_A$): $150 \text{ m}$
Length of Train B ($L_B$): $100 \text{ m}$
The total distance the trains must cover relative to each other to completely pass is the initial distance between them plus their combined lengths.

$$D_{total} = 240 \text{ m} + 150 \text{ m} + 100 \text{ m} = 490 \text{ m}$$

Since the trains are travelling towards each other, their relative speed is the sum of their individual speeds.

$$V_{relative} = V_A + V_B = \frac{410}{9} + \frac{220}{9} = \frac{630}{9} = 70 \text{ m/s}$$

Time ($t$) = Total distance / Relative speed

$$t = \frac{D_{total}}{V_{relative}} = \frac{490 \text{ m}}{70 \text{ m/s}} = 7 \text{ s}$$

The time that elapses before the two trains completely pass each other is $\boxed{\text{7 seconds}}$.