If a person travels in a car at a speed of 36 km/hour, then he will reach his…

Concept

For a fixed distance, time = distance ÷ speed, so the speed a stretch demands is decided by the time still available for it. When a journey must finish on schedule, the scheduled total time is fixed by the reference speed; whatever part of that budget is already spent is gone, and the rest of the distance must be covered inside the time that remains. Required speed = remaining distance ÷ remaining time.

Application

1. Let the whole journey be D km. Covering all of it at 36 km/hour arrives exactly on time, so the scheduled time is T = D/36 hours.

2. Half the journey, that is D/2 km, is completed in (4/5)T.

3. Remaining time = T − (4/5)T = T/5.

4. Remaining distance = D − D/2 = D/2.

5. Required speed = (D/2) ÷ (T/5) = 5D/(2T).

6. Substitute T = D/36: 5D ÷ (2 × D/36) = (5 × 36)/2 = 90 km/hour. D cancels, so the required speed does not depend on how long the journey actually is.

Cross-check

• Take D = 180 km, so T = 180/36 = 5 hours.

• The first 90 km is covered in (4/5) × 5 = 4 hours, leaving 5 − 4 = 1 hour.

• The remaining 90 km in 1 hour is 90 km/hour, which matches the general result.

• Ratio check: the second half now has 1/5 of the total time instead of the 1/2 it was scheduled for, so the speed must rise by the factor (1/2) ÷ (1/5) = 5/2, and 36 × 5/2 = 90 km/hour.