A car travels from $P$ to $Q$ at a constant speed. If its speed were increased by $10$ km/hr , it would have taken one hour lesser to cover the distance. It would have taken further $45$ minutes lesser if the speed was further increased by $10$ km/hr. What is the distance between the two cities?

Using the formula for distance D when equal time differences are involved, the shortest path is found by analyzing the speed inverses, but working directly from the options shows that option A yields the stated conditions. For D = 420 km at the original speed of 60 km/hr, the time taken is 7 hours. Increasing the speed by 10 km/hr to 70 km/hr reduces the time by exactly 1 hour to 6 hours, and further increasing it by 10 km/hr to 80 km/hr reduces the time by another 45 minutes to 5 hr 15 min. Therefore, the result is 420 km.

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