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Explanation:

The speed of the car in the fourth gear is five times its speed in the first gear
⇒ Let the speed in 1^st gear be x kmph, hence speed in 4^th gear will be 5x kmph.

The car takes twice the time to travel a certain distance in the second gear as compared to the third gear
⇒ Speed in 3^rd gear is twice the speed in 2^nd gear.
⇒ Let speeds in 2^nd and 3^rd gears be y & 2y kmph.

In a 100 km journey, if Vinita travels equal distances in each of the gears, she takes 585 minutes
⇒ 25/x + 25/5x + 25/y + 25/2y = 585/60
⇒ 25/x + 5/x + 50/2y + 25/2y = 39/4
⇒ 30/x + 75/2y = 39/4
⇒ 60/x + 75/y = 39/2
⇒ 20/x + 25/y = 13/2
⇒ 5(4/x + 5/y) = 13/2
⇒ 4/x + 5/y = 13/10  ...(1)

Now, we need to find the time taken if Vinita travels 4, 4, 32, 60 kms at 1^st, 2^nd, 3^rd & 4^th gear respectively.
⇒ 4/x + 4/y + 32/2y + 60/5x
= 4/x + 4/y + 16/y + 12/x
= 16/x + 20/y
= 4(4/x + 5/y)
= 4 × 13/10
= 5.2 hours
= 5.2 × 60 minutes
= 312 minutes

Hence, 312.