Aashutosh can cover a certain distance in 84 min by covering 2/3rd of distance at 4 km/h and the rest at 5 km/h. Find the total distance.

Let total distance = x km
According to the question,
x/3 - x/4 = 1/2;
? x/12 = 1/2
? x = 6 km

Length of the bridge = Distance travelled by the person in 4 min = Speed x Time
Speed = 54 km/h = 54 x (5/18) = 3 x 5 = 15 m/s
Time = 4 min = 4 x 60 = 240 s
? Required length = 15 x 240 = 3600 m

Let speed of Kapil in still water = x
and speed of the current= y
? Speed upstream = (x - y)
and speed downstream = (x + y)

According to the question,
12/(x - y) - 12/(x + y) = 6
? 6(x² - y²) = 24y
? x² - y² = 4y
? x² = 4y + y² ....(i)

Again, 12/(2x - y) - 12/(2x + y) = 1
? 4x² - y² = 24y
? x² = (24y + y²)/4 ....(ii)

From Eqs. (i) and (ii), we get
4y + y² = (24y + y²)/4
? 16y +4y² = 24y + y²
? 3y² = 8y
? y = 8/3 = 2²/₃ mile/h

Clearly, motorboat takes 2 h go from Mumbai to Goa and 4 h to go from Goa to Mumbai. i.e.,
Time taken to row upstream = 2 x Time taken to row downstream
? n = 2
Let the speed of stream be V km/h.
According to the formula,
Speed of boat = V(n + 1)/(n - 1)
? 30 = V(2 + 1)/(2 - 1)
? V = 30/3 = 10 km/h

Let the speed of boat in still water be V km/h. Then,
35/(V - 1) + 35/(V + 1) = 12
? 35(V + 1 + V - 1) = 12(V² - 1)
? 6V² - 35V - 6 = 0
? (V - 6)(6V + 1) = 0
? V = 6 (V = - 1/6 is not possible)
So, speed of boat in still water is 6 km/h.

Total distance to covered in 10 h = 80 km
But it covers 40 km in 3/5th of time, i.e, 40 km in 6 h.
? Required time = 10 - 6 = 4 h
And remaining distance = 40 km
Thus, required speed = 40/4 =10 km/h

According to the formula,
Average speed = 2AB/(A + B)
Here, A = 5 km/h, and B = 2 km/h
? Required average speed = (2 x 5 x 2)/7 = 20/7
= 2⁶/₇ km/h

Given that, W + D = 6 ...(i)
[ w = Time taken while walking and
D = Time taken while driving ]
From Eq. (i)
5 +D = 6
? D = 1
2D = 2 x 1 = 2
? He will take 2 h to drive both ways.

Given, D = 84 km, a = 12 km/h and b = 16 km/ h
According to the formula
Distance traveled by A = PR = 2D x a/(a + b)
= 2 x 84 x 12/(12 + 16) = (2 x 84 x 12)/28
= 2 x 6 x 6 = 72 km

Relative speed of policeman = ( 20 - 16 ) x 5/18 = 10/9 m/s
To catch the thief, the policeman in has to gain 200 m = 200 x 9/10 = 180 s
Actual distance covered by policeman in 180 s = 180 x 50/9 = 100 m
? Distance covered by the thief = 1000 - 200 = 800m