Question 1164834
rate * time = distance.
time = distance / rate
rate = distance / time
rate is the distance traveled in a specified period of time.
that's the same as speed in this problem.


the car took 5  hours to travel from town A to town B.
the speed of the car was 90 kilometers per hour.
the distance was therefore 5 * 90 = 450 kilometers.


the bus took 9 hours to travel the same distance.
the speed of the bus was therefore 450/9 = 50 kilometers per hour.


you want the bus to go from town A to town B 1 hour and 36 minutes earlier than its scheduled time.
its scheduled time was 9 hours.
1 hour and 36 minutes earlier is equal to 1 + 36/60 = 1.6 hours earlier.
9 hour minus 1.6 hours is equal to 7.4 hours.
the new scheduled time would need to be 7.4 hours rather than 9 hours so that the bus would go from town A to town B 1 hours and 36 minutes ahead of schedule.


to go from town A to town B in 7.4 hours, the bus would have to travel at a rate of 450 / 7.4 = 60.81081082 kilometers per hour rounded to 8 decimal digits.
that would be an increase in speed of 10.81081082 kilometers per hour rouned to 8 decimal digits.


if the bus could only increase its speed after 3/5 of the journey was completed, then the bus could only increase speed after 3/5 * 450 = 270 kilometers were completed.
that leaves 180 kilometers in which the bus could increase its speed.


270 kilometers at 50 kilometers takes 270/50 = 5.4 hours.
to complete the whole trip in 7.4 hours, then the bus has 7.4 - 5.4 = 2 hours to complete 180 kilometers.
to do that, the bus would need to travel 180 / 2 = 90 kilometers per hour.
that would be an increase of 90 - 50 = 40 kilometers per hour.


to confirm these values are good. do the following.


450 kilometers completed in 7.4 hours equals a speed of 450 / 7.4 = 60.81081081 kilometers per hour for the whole trip.
that's what we calculated before, so that's good.


450 kilometers at a speed of 50 kilometers per hour for the first 270 kilometers and at a speed of 90 kilometers per hour for the last 180 kilometers would take a total of 270 / 50 =  5.4 hours for the first 270 kilometers plus 180 / 90 = 2 hours for the last 180 kilometers for a total of 7.4 hours to cover the entire 450 kilometer distance.