Boat and stream handwritten notes

 Boat and stream

Boat and stream handwritten notes 

Complete math notes

INTRODUCTION:

Downstream: If the boat is moving in the direction of the stream.
Upstream: If the boat is moving in the direction opposite to the direction of the stream.

Assume the speed of the boat in still water as: ‘b’ kilometres per hour (kmph).

Take the speed of the stream as:  ‘s’ kmph.

Hence Aggregate Downstream Speed = b + s kmph. (Boat is moving with the steam of water).

Upstream Net Speed = b - s km/hr. (Boat is moving against the direction of the stream).

Speed of Boat in Still Water:   ½ (Downward Speed + Upward Speed) 

Proof of Basic Formula                 

½ (Downstream Speed + Upstream Speed) = ½ [b + s + (b-s)] = ½ (2b) = b 

(Proved as per the assumption) 

Speed of Stream:  ½ (Downward Speed – Upward Speed)

Proof of Basic Formula 

½ (Downward Speed – Upward Speed) = ½ [b + s – (b-s)] = ½ [b + s - b + s] = ½ (2 s} = s 

(Proved as per the assumption).

 ALWAYS REMEMBER:

• Speed of boat is always greater than the speed of the stream. 
• Downstream speed is always greater than the upward speed.

EXAMPLES:

Example 1. 

A man can row a boat @ 9 kmph in still water. He takes double the time to move upstream than to move the downstream – the same distance. Find the speed of the stream.

Solution:

ATQ (According to Question) and formula given above:
→Let the downward time = 1 hour and so the upward time = 2 hours.
→1/9+s = 2/9-s (Since distance is the same)
→18 + 2 s = 9 – s (By cross multiplication)
→18 – 9 = s + 2 s
→9 = 3 s
Hence s or Speed of stream = 9/3 = 3 kmph Answer.

OR
Simply
→b + s = 2(b –s)
→i.e. b + s = 2b – 2s
→i.e. s + 2s = 2b –b
Or
→b = 3s or 9 = 3s (b = 9 is given) = 3 kmph Answer

Example 2.

A boat runs at 20 kmph along the stream and 10 kmph against the stream. Find the ratio of speed of the boat in still water to that of the speed of the stream.

Solution:

→ ATQ (According to Question) and formula given above:
→ Speed of Boat = ½ (20 + 10) = 15 kmph.
→ Speed of Stream = ½ (20 – 10) = 5 kmph.
→ Ratio: 15:5 = 3:1 Answer.

Example 3.

Find the time taken by the boatman to row 4 kilometres downstream and return to his starting point, if the speed or rate of stream is 2 kilometres per hour and the speed of the boat is 6 kilometres per hour.

Solution: 

ATQ (According to Question) and formula given above:
→ Time = Distance/Speed
→ Time taken = 4/6+2 + 4/6-2 = 4/8 + 4/4 = 0.5 + 1 = 1.5 Hour.

Example 4.

If the speed of the stream is 2 km per hour, and the speed of the boat in still waters is 10 km per hour then find the time taken to cover 60 kms downstream.

Solution: 

→ ATQ (According to Question) and formula given above:
→ 60/10+2 = 60/12 = 5 hours Answer.

Example 5.

Find the speed of the stream when a boat takes 5 hours to travel 60 kms downstream at a rate of 10 kms per hour in still water.

Solution: 

ATQ (According to Question) and formula given above:
→ Speed b + s = 60/5 = 12 kmph
→ Speed b = 10 kmph
→ So, speed s = 12-10 = 2 kmph Answer.

Example 6.

A boat covers a certain distance in one hour downstream with the speed of 10 kmph in still water and the speed of current is 4 kmph. Then find out the distance travelled.

Solution: 

ATQ (According to Question) and formula given above:
→ Distance = Speed x Time = 1 x (10+4) = 14 kms. Answer

Example 7.
A boat takes 6 hours to cover 36 km downstream and 8 hours to cover 32 km upstream. Then the speed of the boat in still water is?

Solution: 

ATQ (According to Question) and formula given above:
→ Speed of Boat = ½ (36/6 + 32/8) = ½ (6+4) = ½(10) = 5 kmph Answer.

Example 8.

A boat takes 6 hours to cover 36 km downstream and 8 hours to cover 32 km upstream. Then the speed of the stream is?

Solution: 

ATQ (According to Question) and formula given above:
→ Speed of Stream ½ (36/6 – 32/8) = ½ (6-4) = ½ (2) = 1 kmph Answer.

Example 9.

If a man rows 6 km downstream in 3 hours and 2 km upstream in 2 hours then how long will he take to cover 9 kms in stationary (still) water?

Solution: 

ATQ (According to Question) and formula given above:
→ Speed of Boat in still waters = ½ (6/3 + 2/2) = ½ (2 + 1) = 1.5 kmph
→ Time taken for 9 kms = 9/1.5 = 6 hours Answer