The chapter of boat and stream is based on the application of time and distance. There are two terms which are frequently used in this chapter are downstream ad upstream.

- Speed downstream = (\( u + v \)) km/hr

Speed upstream = (\( u – v \)) km/hr

**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

- Speed in still water = \( \frac{1}{2}(a + b) \) km/hr

Rate of stream = \( \frac{1}{2}(a – b) \) 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 = \(\frac{1}{2}\)(10 + 7) km/hr = 8.5 km/hr.

Rate of current = \(\frac{1}{2}\)(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 = \(\frac{1}{2}\) (Downstream Speed + Upstream Speed)

= \(\frac{1}{2}\) (18+12)

= 15 km/hr

Speed of the current = \(\frac{1}{2}\) (Downstream Speed – Upstream Speed)

= \(\frac{1}{2}\) (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) = \(\frac{28}{4}\) = 7 km/hr

Upstream Speed (v) = \(\frac{12}{3}\) = 4 km/hr

Speed of the boat or swimmer in still water = \(\frac{1}{2}\) (Downstream Speed + Upstream Speed)

= \(\frac{1}{2}\) (7+4)

= 5.5 km/hr

Speed of the current = \(\frac{1}{2}\) (Downstream Speed – Upstream Speed)

= \(\frac{1}{2}\) (7-4)

= 1.5 km/hr

- Speed downstream = (\( u + v \)) km/hr

Speed upstream = (\( u – v \)) km/hr

2. If the speed downstream is \( a \) km/hr and the speed upstream is \( b \) km/hr. Then,

- Speed in still water = \( \frac{1}{2}(a + b) \) km/hr

Rate of stream = \( \frac{1}{2}(a – b) \) km/hr

- 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.