A person riding a bike crosses a bridge with a speed of 54 km/h. What is the length of the bridge, if he takes 4 min to cross the bridge?

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.

Let the distance between the two ports = L km
Then, speed downstream = L/4
and speed upstream = L/5
? Speed of the stream = 1/2(L/4 - L/5)
? 1/2[(5L - 4L)/20] = 2
? L = 80 km

Speed upstream + Speed downstream = 2 x 7¹/₂ = 15 km/h
Since, the time taken is in the ratio 2 : 1
hence the speed will be in the ratio 1 : 2
? Speed upstream = (1/3) x 15 = 5 km/h
and speed downstream = (2/3) x 15 = 10 km/h
? Speed of stream = (10 - 5)/2 = 2.5 km/h

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

Let the total distance = L
Then, according to the question,
(2L/3)/ 4 + (1 - 2/3)L/5 = 84/60
? 2L/12 + L/15 = 84/60
? L/6 + L/15 = 84/60
? 5L + 2L = 42
? 7L = 42
? L = 42/7 = 6 km

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.