A steamer 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, then find the distance between the two ports.?

Speed upstream = 10 / (30/60) = (10 x 60)/30 = 20 km/h
Speed downstream = 10/(25/60) = (10 x 60)/25 = 24 km/h
? Speed of the river's current = (24 - 20)/2 = 4/2 = 2 km/h

Let velocity of the stream be x km/h
? Velocity of the boat downstream = (4 + x) km/h
and velocity of the boat upstream = (4 - x) km/h
According to the question, 3(4 + x) = 9(4 - x)
? 12 + 3x = 36 - 9x
? 12x = 24
? x = 2 km/h

Let the distance be D km.
Speed download = (5 +2) = 7 km/h
and speed upstream = (5 -2) = 3 km/h
According to the question,
D/3 - D/7 = 2
? 7D - 3D = 21 x 2
? D = (21 x 2)/4 = 10.5 km

Let speed of stream be x km/h
Given speed of motorboat in still water = 45 km/h
? Speed of boat along stream = (45 + x) km/h

According to the question,
45 + x = 80 x 1¹/₃
? 45 + x = 80 x 3 / 4
? x = 60 - 45 = 15
? Speed of boat against stream = 45 - 15 = 30 km/h
Hence, required time = Distance / Speed = 80 / 30
= (8 x 60)/3 = 160 min
= 2 h 40 min

Let x be the speed of the boat and y be the speed of the current.
Speed upstream = (x - y) km/h
Speed downstream = (x + y) km/h
According to the question,
20 /(x - y) + 20/(x + y) = 25/36
In this equation, there are two variables but only one equation. So, the value of x cannot be determined.

For the first boat,
Speed of stream : Speed of boat = 2: 5
Let speed of stream be 2x km/h and speed of boat = 5x km/h

Similarly, for the second boat
Speed of stream be 3y km/h and speed of boat = 4y

In both of the conditions, river is same.
? 2x = 3y
? x = 3y/2
Thus, required ratio in speeds of boats in still water
= Thus, required ratio in speeds of boats in still water
= 5x : 4y = (5 x 3)y/2 : 4y = 15 : 8

Let the speed of man and current be U and V km/h, respectively.
Speed upstream = (U - V) km/h
Speed down stream = (U + V) km/h
According to the question,
(3 x 60)/(4 x 15) = U - V
? U - V = 3 ......(i)

and (3/4) x (60/10) = U + V
? U + V = 9/2 ...(ii)

On adding Eqs. (i) and (ii), we get
2U = 3 + 9/2
? U = 15/4

On putting the value of U in Eq. (ii), we get
15/4 + V = 9/2
V = 9/2 - 15/4 = 18 - 15/4
? V = 3/4

Hence, speed of man U = 15/4 and
Speed of current V = 3/4
Hence, required ratio = 15/4 : 3/4 = 5 : 1

Let boat's rate upstream be U and boat's rate downstream = V

According to the question.
Distance covered in 528 min = Distance covered in 240 min
? Distance covered in 8 h 48 min = Distance covered in 4 h
? U x 8⁴/₅ = V x 4
? (44 x U)/5 = 4V
? V = (11 x U)/5
? Required ratio = (V + U)/2 : (V - U)/2
? = (11U/5 + U)/2 : (11U/5 - U)/ 2
? = 16U/5 : 6U/5 = 8 : 3

Let the speed of a boat and stream be U and V km/h
? Speed of boat along stream = (U + V) km/h
and speed of boat against stream = (U - V) km/h

According to the question,
25/(U - V) + 39/(U + V) = 8 ....(i)
and 35/(U - V) + 52/(U + V ) = 11 ..(ii)

On multiplying Eq. (i) by 4 and Eq. (ii) by 3, then subtract Eq. (ii) from Eq, we get
100/(U - V) - 105/(U - V) = -1
? 5/(U - V) = 1
? U - V = 5 ...(iii)

On substituting the value of (U - V) = 5 in Eq. (i) we get
25/5 + 39/(U + V) = 8
? 39/(U + V) = 8 - 5
? U + V = 39/3
? U + V = 13 ....(iv)

On solving Eqs. (iii) and (iv), we get
U = 9 and V = 4
Hence, speed of stream = 4 km/h

Upstream distance = (4 - 2) x 9 = 18 km
? Required time = 18/(4 + 2) = 3 hrs.