A streamer goes downstream from one port to another in 4 h. It covers the same distance upstream in 5 h. If the speed of the stream is 2 km/h , find the distance between the two ports?

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

Relative speed of policeman with thief = (10 - 8) = 2 km/h
Now, time taken to cover 100 m with speed of 2 km/h or 2000 m/h

Now, Speed = Distance / Time
? Time = D/S = 100/2000
= 1/20 h or 60/20 min = 3 min

Let the distance be D km
Difference in timings = 8 min = 8/60 h = 2/15 h
? D/3 - D/4 = 2/15
? D/12 = 2/15
? D = 2 x 12/15 = 1.6 km

New speed = 8/11 of usual speed
New time = 11/8 of usual time
? 11/8 of usual time = 11/2 h
? Usual time = (11 x 8)/(2 x 11) = 4 h
? Time saved = 5¹/₂- 4
= 1¹/₂

Average speed = [(30/60) x 40 + (45/60) x 60 + (2 x 70)] / [(30/60) + (45/60) + 2]
= (20 + 45 + 140)/[(30 + 45 + 120)/60]
= (205/195) x 60 km/h
= 63.07 km/h
= 63 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.

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 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

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