Boats and Streams

Boat and Stream refers to a category of problems that analyze the movement of a boat in a river or water body with a current. 

To excel in boat and stream problems, it's essential to understand key concepts such as:

• Stream: Moving water in a river or any other water body.
• Upstream: Moving against the direction of the stream or current.
• Downstream: Moving along the direction of the stream or current.
• Still Water: Water in a river or any other water body that is not flowing or stationary.

Upstream and Downstream Formulas

Upstream and downstream are important concepts in boat and stream problems, and understanding the relevant formulas is crucial for solving related questions.

Solved Examples - Boats and Streams

Question 1: A boat travels 10 km downstream in 2 hours and 5 km upstream in 2 hours.
Find:

1. The boat's speed in still water.
2. The speed of the stream.

Solution:

Downstream Speed (with current): Speed=Distance/Time=10 km/2 hrs=5 km/h

Upstream Speed (against current): Speed=5 km/2 hrs=2.5 km/h

Let: Then: b+s=5 (Downstream)

b−s =2.5 (Upstream)

Add the two equations: (b+s)+(b−s)=5+2.5

2b = 7.5  ⟹  b = 3.75 km/h

Find s: s=5−b=5−3.75=1.25 km/h

Final Answer:

• Boat speed in still water = 3.75 km/h
• Stream speed = 1.25 km/h

Question 2: A boatman can row a boat upstream at 10 km/hr and downstream at 16 km/hr. Find the speed of the boat in still water and the speed of the stream.

Solution:

We are given that speed downstream, D = 16 km/hr and speed upstream, U = 10 km/hr.

Therefore, the speed of the boat in still water = 0.5 × (D + U) km/hr = 0.5 × (16 + 10) = 13 km/hr.

Also, the speed of the stream = 0.5 × (D – U) km/hr = 0.5 × (16 – 10) = 3 km/hr.

Another method:

Speed of the stream = 0.5 × (D – U) = 0.5 × 6 = 3 km/hr.

Speed of the boat in still water = Speed of the stream + Speed Upstream = 3 + 10 = 13 km/hr.

Question 3: A boat’s speed in still water is 10 km/hr and the speed of the stream is 2 km/hr. Find the downstream and upstream speeds.

Solution:

Downstream = B + S = 10 + 2 = 12 km/hr
Upstream = B − S = 10 − 2 = 8 km/hr

Question 4: A boat covers 24 km downstream in 3 hours and the same distance upstream in 4 hours.
Find:

• Speed of the boat in still water
• Speed of the stream

Solution:

Downstream speed = 24 ÷ 3 = 8 km/hr
Upstream speed = 24 ÷ 4 = 6 km/hr

Boat speed = ½ (8 + 6) = 7 km/hr
Stream speed = ½ (8 − 6) = 1 km/hr

Question 5: A boat travels 30 km downstream and returns back upstream in a total time of 10 hours. The speed of the stream is 2 km/hr. Find the speed of the boat in still water.

Solution:

Let boat speed = b

Downstream speed = b + 2
Upstream speed = b − 2

Total time:

\frac{30}{b+2} + \frac{30}{b-2} = 10

Solve:

\frac{30(b-2) + 30(b+2)}{b^2 - 4} = 10

\frac{60b}{b^2 - 4} = 10

60b=10(b^2−4)

6b = b^2 - 4 \Rightarrow b^2 - 6b - 4 = 0

b = 3 \pm \sqrt{13}

Take positive:
b ≈ 3 + 3.6 = 6.6 km/hr (approx)

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