# Boats – Stream Problems

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# Boats – Stream Problems

### Introduction

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.

### Methods

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 â€“ y) km/h = (8 â€“ 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 = $$\frac{1}{2}$$(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

### Formulae

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

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

### Samples

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

â‡’ 12 $$x$$ + 16 $$x$$ = 14

â‡’ $$x$$ = $$\frac{1}{2}$$

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

â‡’ $$\frac{3x}{2}$$kmph = 12

â‡’ $$x$$ = 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, $$\frac{6}{x + 2} + \frac{6}{x – 2}$$ = $$\frac{33}{60}$$

â‡’ 11 $$x^2 – 240x$$ – 44 = 0

â‡’ 11 $$x^2 – 242x +2x$$ – 44 = 0

â‡’ ( $$x$$ – 22 )( $$11x$$ + 2 ) = 0

â‡’ x = 22

Hence, Speed of motor boat in still water = 22 kmph.