Downstream : In water, the direction along the stream is downstream.
Upstream : In water, the direction against the stream is upstream.
Example 1: A man row with a speed of 8 km/h in still water. Find the downstream and upstream speed of boat, if the speed of stream is 4 km/h.?
Solution:

Downstream speed = (u + v) km/h = (8 + 4) = 12 km/h
Upstream speed = (x – y) km/h = (8 – 4) = 4 km/h

Example 2: Speed of boat in still water is 16 km/hr. If the speed of the stream is 4 km/hr, find its downstream and upstream speeds.
Solution:

Downstream Speed = (u + v) km/h = 16 + 4 = 20 km/hr
Upstream Speed = (u - v) km/h = 16 - 4 = 12 km/hr

Example 1: A man can row upstream at 7 kmph and downstream at 10 kmph. Find man's rate in still water and the rate of current.
Solution:

Rate in still water = [latex]\frac{1}{2}[/latex](10 + 7) km/hr = 8.5 km/hr.
Rate of current = [latex]\frac{1}{2}[/latex](10 - 7) km/hr = 1.5 km/hr.

Example 2: A man can row downstream at 18 km/hr and upstream at 12 km/hr. Find his speed in still water and the rate of the current.
Solution:

Speed of the boat or swimmer in still water = [latex]\frac{1}{2}[/latex] (Downstream Speed + Upstream Speed)
= [latex]\frac{1}{2}[/latex] (18+12)
= 15 km/hr
Speed of the current = [latex]\frac{1}{2}[/latex] (Downstream Speed - Upstream Speed)
= [latex]\frac{1}{2}[/latex] (18-12)
= 3 km/hr

Example 3: A man swims downstream 28 km in 4 hrs and upstream 12 km in 3 hrs. Find his speed in still water and also the speed of the current.
Solution:

Downstream Speed (u) = [latex]\frac{28}{4}[/latex] = 7 km/hr
Upstream Speed (v) = [latex]\frac{12}{3}[/latex] = 4 km/hr
Speed of the boat or swimmer in still water = [latex]\frac{1}{2}[/latex] (Downstream Speed + Upstream Speed)
= [latex]\frac{1}{2}[/latex] (7+4)
= 5.5 km/hr
Speed of the current = [latex]\frac{1}{2}[/latex] (Downstream Speed - Upstream Speed)
= [latex]\frac{1}{2}[/latex] (7-4)
= 1.5 km/hr

1. If the speed of the boat in still water is [latex] u [/latex] km/hr and the speed of the stream is [latex] v [/latex] km/hr. Then,

Speed downstream = ([latex] u + v [/latex]) km/hr
Speed upstream = ([latex] u - v [/latex]) km/hr

2. If the speed downstream is [latex] a [/latex] km/hr and the speed upstream is [latex] b [/latex] km/hr. Then,

Speed in still water = [latex] \frac{1}{2}(a + b) [/latex] km/hr
Rate of stream = [latex] \frac{1}{2}(a - b) [/latex] km/hr

1. A man rows to place 48 km distant and back in 14 hours. Finds that the man can row 4 km with the stream in the same time as 3 km against the stream. Find the rate of the stream?
Solution:

Let the man moves 4 km downstream in \( x \) hours.
Then in speed downstream = km/hr
Speed in upstream = \( \frac{4}{x} \) km/hr
Therefore, \( \frac{48}{\frac{4}{x}} + \frac{48}{\frac{3}{x}} \) = 14
⇒ 12 \( x \) + 16 \( x \) = 14
⇒ \( x \) = \( \frac{1}{2} \)
Then, speed in downstream = 8 km/hr
speed in upstream = 6 km/hr
Therefore, Rate of stream = \( \frac{1}{2}(8 - 6)\) = 1 km/hr

2. A man can row downstream at the rate of 14 km/hr and upstream at 5 km/hr. Find man's rate in still water and the rate of current?
Solution:

Given that
Rate of downstream = 14 kmph
Rate of upstream = 5 kmph
Then,
Rate of still water = \( \frac{1}{2}(14 + 5) \) = 9.5 km/hr
Rate of current = \( \frac{1}{2}(14 - 5) \) = 4.5 km/hr

3. A man can row 12 kmph in still water. It takes twice as long to row down the river. Find the rate of stream ? Solution:

Given that
A man can row 12 kmph in still water.
Let the man's rate upstream \( x \) kmph
Then, rate downstream = 2 \( x \) kmph
Therefore, Rate in still water = \( \frac{1}{2}(2x + x) \) = \( \frac{3x}{2} \)kmph
⇒ \( \frac{3x}{2} \)kmph = 12
⇒ \( x \) = 8
Rate of upstream = 8 kmph
Rate of down stream = 16 kmph
Therefore, Rate of stream = \( \frac{1}{2}(16 - 8) \) = 4 kmph

4. There is a road besides a river. Two friends started from a place A, moved to school situated at another place B and then returned to A again. One of them moves on a cycle at a speed of 14 kmph, while the other sails on a boat at a speed of 10 kmph. If the river flows at the speed of 10 kmph. If the river flows at the speed of 2 kmph, which of two friends will return to place A first?
Solution:

Given,
The cyclist moves both ways at a speed of 12 kmph.
So, average speed of the cyclist = 12 kmph.
The boat sailor moves downstream at (10 + 4) i.e. 14 kmph and upstream at (10 - 4) i.e. 6 kmph.
So, average speed of the boat sailor = \( \frac{2 * 14 * 6}{14 + 6} \) kmph = \( \frac{42}{5} \) kmph = 8.4 kmph
Since, the average speed of the cyclist is greater, the man will return to A first.

5. In a stream running at 5 kmph, a motor boat goes 8 kmph upstream and back again to the starting point in 22 minutes. Find the speed of the motor boat in still water?
Solution:

Let the speed of the motor boat in still water be \( x \) kmph.
Then, Speed downstream = ( \( x + 2\) ) kmph
Speed upstream = ( \( x - 2\) ) kmph
Therefore, \( \frac{6}{x + 2} + \frac{6}{x - 2} \) = \( \frac{33}{60} \)
⇒ 11 \( x^2 - 240x\) - 44 = 0
⇒ 11 \( x^2 - 242x +2x \) - 44 = 0
⇒ ( \( x \) - 22 )( \( 11x \) + 2 ) = 0
⇒ x = 22
Hence, Speed of motor boat in still water = 22 kmph.