there are two cities P and Q separated by 60 km, two buses A and B start from P and Q respectively at 6AM and stop at 6PM how many times do they cross

Question

(assume that the buses travel continuously at constant speeds, without any stopping or turning time at the ends.) I figured out the time it takes to cross once (6/7 hours) but after crossing once things get complicated

Answer 1

If they cross once after 6/7 hours, then they should be halfway between cities at that point. It would take twice as long, 12/7 hours, for them to reach opposite cities and turn around. Next you take your total time allotment (12 hours) and divide by 12/7. $$\frac{12}{\frac{12}{7}} = 7$$

They cross 7 times. Once for each time they switch cities.

Answer 2

Since we now know that bus A runs at 25km/h and bus B at 45 km/h we should find their total distance travelled in 12 hrs.

Bus A: $12\times25=300 km$

Bus B: $12\times45=540 km$

Logically, we can see that the number of times the fastest bus goes from city to city completely will give us one crossing of buses, as the slow one cannot catch up to it, but must be somewhere in the interval [P,Q]. We Therefore divide 540 (the total distance travelled of the fast bus) by 60 (distance between cities) and get the number of crossings:

$$\frac{540}{60}=9$$

They cross 9 times.