## Boats and Streams

"Boats and streams" is an application of concepts of speed, time and distance. Speed of river either aides a swimmer/boat, while traveling with the direction of river or it apposes when traveling against the direction of river.

**Still Water:** If the speed of water of a river is zero, then water is considered to be still water.

**Stream Water:** If the water of a river is moving at a certain speed, then it is called as stream water.

**Speed of Boat: **Speed of boat means speed of boat in still water

**Upstream:** If a boat or a swimmer moves in the opposite direction of the stream, then it is called upstream.

**Downstream:** If a boat or a swimmer moves in the same direction of the stream, them it is called downstream.

#### Basic Rule of Boats and Streams

**Rule 1: **If the speed of a boat or swimmer in still water is x km/h and speed of the stream is y km/h, then

Speed of boat or swimmer in downstream = ( x + y ) km/h |

**Example**: A man can row with a speed of 8 km/h in still water. If the speed of stream is 3 km/h, what will be his speed with the stream.
**Solution:** Given, speed of man in still water = 8 km/h

speed of stream = 3 km/h

∴ downstream speed = ( x + y )

= 8 + 3

= 11 km/h

**Rule 2: **If the speed of a boat or swimmer in still water is x km/h and speed of the stream is y km/h, then

Speed of boat or swimmer in upstream = ( x - y ) km/h |

**Example: **If the speed of a boat in still water is 5 km/h and the rate of stream is 2 km/h, then find upstream speed of the boat.
**Solution:** Given, speed of boat = 5 km/h

speed of stream = 2 km/h

∴ upstream speed = x - y

= 5 - 2

= 3 km/h

**Rule 3:** If the speed of a boat in downstream is x km/h and speed of the boat in upstream is y km/h, then

Speed of boat in still water = | 1 | (speed in downstream + speed in upstream) |

2 |

**Example: **A man can row a boat in downstream at 14 km/h and in upstream at 10 km/h. Find the speed of the boat that the man can row in still water. **Solution: **Given, speed in upstream = 10 km/h

speed in downstream = 14 km/h

speed of boat in still water = | 1 | (speed in downstream + speed in upstream) |

2 |

speed of boat in still water = | 1 | (14 + 10) |

2 |

speed of boat in still water = | 1 | (24) |

2 |

= 12 km/h

**Rule 4:** If the speed of a boat in downstream is x km/h and speed of the boat in upstream is y km/h, then

Speed of stream = | 1 | (speed in downstream - speed in upstream) |

2 |

**Example: ** Ram can row upstream at 11 km/h and downstream at 17 km/h. Find the rate of the current.
**Solution:-** Given, speed in upstream = 11 km/h

Speed in downstream = 17 km/h

Speed of stream = | 1 | (speed in downstream - speed in upstream) |

2 |

Speed of stream = | 1 | (17 - 11) |

2 |

Speed of stream = | 1 | (6) |

2 |

= 3 km/h

**Example**: A boat covers 56 km in 8 hours in downstream motion. And return to the same place in 14 hours. Find the speed of boat and Stream.
**Solution:**

speed = | distance |

time |

speed in downstream = | 56 | = 7 km/h |

8 |

speed in upstream = | 56 | = 4 km/h |

14 |

speed of boat = | 1 | (speed in downstream + speed in upstream) |

2 |

= | 1 | (7 + 4) |

2 |

= | 1 | × 11 |

2 |

= 5.5 km/h

Speed of stream = | 1 | (speed in downstream - speed in upstream) |

2 |

Speed of stream = | 1 | (7 - 4) |

2 |

Speed of stream = | 1 | × 3 |

2 |

= 1.5 km/h

**Rule 5:** If speed of stream is x and a boat takes t times longer to row up as compared to row down the river, then

Speed of boat in still water = x | (t + 1) |

(t - 1) |

**Example:** A can row 10 km/h in still water. It takes his thrice as long as to row up as to row down the river. Find the speed of stream.
**Solution: **Let the rate of current be y km/h.

Speed of A in still water = 10 km/h

Speed in downstream = ( 10 + y ) km/h

Speed in upstream = ( 10 - y ) km/h

Time = | Distance |

Speed |

t = | …..(1) | |

10 + y |

3t = | …..(2) | |

10 - y |

Putting the value of t from Eq.(1) in Eq.(2), we get

Speed of boat in still water = x | |

(t - 1) |

3 × | + y = | - y | ||

10 | 10 |

or, 3 ( 10 - y ) = 10 + y

or, 30 - 3y = 10 + y

or, 30 - 10 = 3y + y

or, 20 = 4y

∴ y = 20/4 = 5

∴ Speed of stream = 5 km/h

**By Rule**,

Here, Speed of A = 10 km/h and t = 3

Speed of boat in still water = x | |

(t - 1) |

or, 10 = x | |

(3 - 1) |

or, 10 = | |

2 |

or, 10 = x × 2

∴ x = 10/2 = 5 km/h

= speed of stream

**Rule** **6:** A man rows a certain distance downstream in t_{1} hour and returns the same distance in t_{2} hour. When the stream flows at the rate of y km/h, then

Speed of the man in still water = y | (t_{1} + t_{2}) |

(t_{2} - t_{1}) |

**Example**: Rohan can row a certain distance downstream in 6 h and can return the same distance in 8 h. If the stream flows at the rate of 4 km/h, then find the speed of Rohan in still water.
**Solution:** Let the speed of Rohan in still water be x km/h.

Speed of stream = 4 km/h

Speed in downstream = ( x + 4 ) km/h

Speed in upstream = ( x - 4 ) km/h

According to the question,

Distance traveled in upstream = Distance traveled in upstream

Distance = Speed × Time ( x - 4 ) × 8 = ( x + 4 ) × 6

or, 8x - 32 = 6x + 24

or, 8x - 6x = 32 + 24

or, 2x = 8 ∴ x = 56/2 = 28

∴ speed of Rohan in still water = 28 km/h
**By Rule** ,

Speed of the man in still water = y | (t_{1} + t_{2}) |

(t_{2} - t_{1}) |

= 4 | (6 + 8) |

(8 - 6) |

= 4 × | (14) |

(2) |

= 4 × 7

= 28 km/h

**Rule 7:** When boat or swimmer's speed in still water is x km/h and river is flowing with a speed of y km/h and time taken to cover a certain distance upstream is t more than the time taken to cover the same distance downstream, then

Distance = t | (x² - y²) |

2y |

**Example: **A boat speed in still water is 8 km/h, while river is flowing with a speed of 4 km/h and time taken to cover a certain distance upstream is 5 hours more than time taken to cover the same distance downstream. Find the distance.

**Solution: **Let the distance be D km

given, speed of boat in still water = 8 km/h

Speed of water = 4 km/h

Speed in downstream = 8 + 4 = 12 km/h

Speed in upstream = 8 - 4 = 4 km/h

According to the question,

D/4 - D/12 = 5

or, 3D - D /12 = 5

or, 2D/12 = 5

or, 2D = 12 × 5

or, 2D = 60

∴ D = 60/2 = 30 km
**By Rule**,

Here, x = 8 km/h, y = 4 km/h and t = 5 hours

Distance = t | (x² - y²) | (speed in downstream + speed in upstream) |

2y |

= 5 | (8² - 4²) |

2 × 4 |

= 5 | (64 - 16) |

8 |

= 5 × | 48 |

8 |

= 5 × 6

= 30 km

**Rule 8:** If boat or swimmer's speed in still water is x km/h and river is flowing with a speed of y km/h, then average speed in going to a certain place and coming back to

starting point is given by | ( x + y ) ( x - y ) | km/h |

x |

**Example:** A rows in still water with a speed of 7 km/h to go to a certain place and to come back. Find his average speed for the whole journey, if the river is flowing with a speed of 3 km/h.

**Solution:** Given, speed of A in still water = 7 km/h

Speed of stream = 3 km/h

Speed in upstream = 7 - 3 = 4 km/h

Speed in downstream = 7 + 3 = 10 km/h

Let the distance in one direction be D km.

Time taken in upstream = | D |

4 |

Time taken in downstream = | D |

11 |

Average speed = | Total distance |

Total time taken in travel |

= | ||||

+ | ||||

10 | 10 |

= | ||||

40 |

= | |

14D |

= 40/7

= 5.7 km/h

**By Rule**,

Here, x = 7 km/h and y = 3 km/h

Average speed = | km/h | |

x |

= | km/h | |

7 |

= 10 × 4 /7

= 40/7

= 5.7 km/h

**Rule 9:** If a boat covers D km distance in t_{1} hour along the direction of river and it covers same distance in t_{2} hour against the direction of river, then

Speed of boat in still water = | _{2} + t_{1} |
|||

2 | t_{1}.t_{2} |

Speed of the stream = | _{2} - t_{1} |
|||

2 | t_{1}.t_{2} |

**Example**: A boat covers 24 km in 2 hour with downstream and covers the same distance in 4 hour with upstream. Then, find the speed of boat in still water and speed of stream.

**Solution: **Given, Distance = 24 km

Speed = distance/time |

Speed in downstream = | |

2 |

= 12 km/h

Speed in upstream = 24/4 = 6 km/h

∴ Speed of boat in still water = | (speed in downstream + speed in upstream) | |

2 |

= | (12 + 6) | |

2 |

= | × 18 | |

2 |

= 9 km/h

∴ Speed of stream = | (speed in downstream - speed in upstream) | |

2 |

= | (12 - 6) | |

2 |

= | × 6 | |

2 |

= 3 km/h

**By Rule**,

Here, D = 24 km,

t

_{1}= 2 hour and

t

_{2}= 4 hour

∴ Speed of boat in still water = | _{2} + t_{1} |
|||

2 | t_{1}.t_{2} |

= | ||||

2 | 2 × 4 |

= 12 × | |

8 |

= 12 × | |

4 |

= 9 km/h

Speed of the stream = | _{2} - t_{1} |
|||

2 | t_{1}.t_{2} |

= | ||||

2 | 2 × 4 |

= 12 × | |

8 |

= 3 km/h