Boat and Stream

1. BASIC CONCEPT

Boats and Streams problems are applications of speed, time and distance. The special feature is that the speed of a boat changes according to the direction in which it moves relative to the flowing water.

``` Important Terms

Boat speed in still water: Speed of the boat when there is no current.
Stream speed: Speed at which the water is flowing.
Downstream: Boat moves in the same direction as the stream.
Upstream: Boat moves against the direction of the stream.

```

2. FUNDAMENTAL FORMULAS

``` Let:
B = Speed of boat in still water
S = Speed of stream

Downstream speed = B + S
Upstream speed = B − S ```

``` Boat speed in still water:

B = (Downstream speed + Upstream speed) ÷ 2

Stream speed:

S = (Downstream speed − Upstream speed) ÷ 2 ```

3. DOWNSTREAM MOTION

When the boat moves in the same direction as the stream, the stream helps the boat. Therefore, the speeds are added.

``` Downstream Speed = Boat Speed + Stream Speed

Distance = Downstream Speed × Time
Time = Distance ÷ Downstream Speed ```

``` Example 1: A boat moves at 12 km/h in still water. The stream flows at 3 km/h. Find the downstream speed.

Downstream speed = 12 + 3
= 15 km/h. ```

4. UPSTREAM MOTION

When the boat moves against the stream, the current opposes the motion. Therefore, the stream speed is subtracted.

``` Upstream Speed = Boat Speed − Stream Speed ```

``` Example 2: A boat travels at 15 km/h in still water and the stream speed is 4 km/h.

Upstream speed = 15 − 4
= 11 km/h. ```

5. DIRECT APPLICATION

``` Example 3: A boat travels downstream at 18 km/h. If the stream speed is 5 km/h, find its speed in still water.

Boat speed = 18 − 5
= 13 km/h. ```

``` Remember:
Downstream → Add stream speed
Upstream → Subtract stream speed ```

6. TIME AND DISTANCE

``` Time = Distance ÷ Speed

Distance = Speed × Time

Speed = Distance ÷ Time ```

``` Example 1: A boat travels 60 km downstream at 15 km/h. Find the time taken.

Time = 60/15
= 4 hours. ```

``` Example 2: A boat travels upstream at 10 km/h for 3 hours. Find the distance travelled.

Distance = 10 × 3
= 30 km. ```

7. ROUND TRIP

In a round-trip problem, a boat travels a certain distance downstream and returns the same distance upstream.

``` Total Distance = Downstream Distance + Upstream Distance

Total Time = Downstream Time + Upstream Time

Average Speed = Total Distance ÷ Total Time ```

``` Example 3: A boat covers 30 km downstream at 15 km/h and returns upstream at 10 km/h.

Downstream time = 30/15 = 2 hours
Upstream time = 30/10 = 3 hours
Total time = 2 + 3 = 5 hours

Total distance = 30 + 30 = 60 km
Average speed = 60/5 = 12 km/h. ```

8. AVERAGE SPEED FOR EQUAL DISTANCES

``` If equal distances are travelled at speeds a and b:

Average Speed = 2ab/(a+b) ```

``` Example 4: A boat covers equal distances downstream and upstream at 18 km/h and 12 km/h respectively.

Average speed = 2 × 18 × 12/(18 + 12)
= 432/30
= 14.4 km/h. ```

9. FINDING BOAT AND STREAM SPEED

``` Example 5: The downstream speed of a boat is 20 km/h and the upstream speed is 12 km/h.

Boat speed = (20 + 12)/2 = 16 km/h

Stream speed = (20 − 12)/2 = 4 km/h. ```

10. IMPORTANT RELATION

``` Downstream Speed + Upstream Speed = 2 × Boat Speed

Downstream Speed − Upstream Speed = 2 × Stream Speed ```

``` Shortcut: If downstream and upstream speeds are known, simply take their half-sum for boat speed and half-difference for stream speed. ```

11. EFFECT OF STREAM SPEED

The stream affects the effective speed of a boat. If the boat speed in still water is greater than the stream speed, the boat can move upstream.

``` Downstream = B + S
Upstream = B − S ```

``` Example 1: A boat's speed in still water is 20 km/h and the stream speed is 6 km/h.

Downstream = 20 + 6 = 26 km/h
Upstream = 20 − 6 = 14 km/h. ```

12. TIME DIFFERENCE

Sometimes a boat takes different amounts of time to travel the same distance downstream and upstream.

``` Downstream Time = D/(B+S)

Upstream Time = D/(B−S)

Difference in Time = D/(B−S) − D/(B+S) ```

``` Example 2: A boat travels 60 km. Its speed in still water is 18 km/h and stream speed is 3 km/h.

Downstream speed = 18 + 3 = 21 km/h
Upstream speed = 18 − 3 = 15 km/h

Downstream time = 60/21 = 20/7 h
Upstream time = 60/15 = 4 h

Difference = 4 − 20/7 = 8/7 hours. ```

13. DISTANCE COVERED IN DIFFERENT DIRECTIONS

``` Example 3: A boat travels downstream for 4 hours at 16 km/h. Find the distance.

Distance = 16 × 4 = 64 km. ```

14. SAME DISTANCE — TIME RATIO

When the same distance is covered upstream and downstream, time is inversely proportional to speed.

``` Time Ratio:

Downstream Time : Upstream Time = Upstream Speed : Downstream Speed ```

``` Example 4: Downstream speed is 24 km/h and upstream speed is 16 km/h. Find the ratio of times taken for the same distance.

Time ratio = 16 : 24
= 2 : 3. ```

15. BOAT SPEED AND STREAM SPEED FROM RATIO

``` Example 5: The speeds of a boat downstream and upstream are in the ratio 5:3. Find the ratio of boat speed to stream speed.

Let downstream = 5x
Upstream = 3x

Boat speed = (5x + 3x)/2 = 4x
Stream speed = (5x − 3x)/2 = x

Therefore, Boat : Stream = 4 : 1. ```

16. IMPORTANT FORMULA TABLE

``` ```

Quantity	Formula
Downstream speed	B + S
Upstream speed	B − S
Boat speed	(D + U)/2
Stream speed	(D − U)/2
Downstream time	Distance ÷ (B + S)
Upstream time	Distance ÷ (B − S)
Average speed for equal distances	2DU/(D+U)

``` Notation: B = boat speed in still water
S = stream speed
D = downstream speed
U = upstream speed ```

17. COMPLETE FORMULA SHEET

``` ```

Concept	Formula
Boat speed	(Downstream + Upstream)/2
Stream speed	(Downstream − Upstream)/2
Downstream speed	B + S
Upstream speed	B − S
Distance	Speed × Time
Time	Distance ÷ Speed
Average speed	Total distance ÷ Total time
Equal-distance average	2ab/(a+b)
Same-distance time ratio	Inverse of speed ratio

18. SOLVED REVISION EXAMPLES

``` Example 1: A boat's speed in still water is 15 km/h and stream speed is 5 km/h.

Downstream = 15 + 5 = 20 km/h
Upstream = 15 − 5 = 10 km/h. ```

``` Example 2: A boat travels 40 km downstream at 20 km/h and returns at 10 km/h.

Downstream time = 40/20 = 2 h
Upstream time = 40/10 = 4 h
Total time = 6 h
Average speed = 80/6 = 13⅓ km/h. ```

``` Example 3: A boat travels downstream at 24 km/h and upstream at 16 km/h.

Boat speed = (24 + 16)/2 = 20 km/h
Stream speed = (24 − 16)/2 = 4 km/h. ```

``` Example 4: A boat covers 75 km upstream at 15 km/h. Find the time.

Time = 75/15 = 5 hours. ```

19. PRACTICE QUESTIONS

A boat moves at 18 km/h in still water and the stream speed is 4 km/h. Find downstream and upstream speeds.
The downstream speed of a boat is 22 km/h and upstream speed is 14 km/h. Find boat speed and stream speed.
A boat travels 60 km downstream at 20 km/h. Find the time.
A boat travels 48 km upstream at 12 km/h. Find the time.
A boat travels 30 km downstream at 15 km/h and returns at 10 km/h. Find total time.
A boat covers equal distances at 20 km/h and 12 km/h. Find its average speed.
The downstream and upstream speeds are 25 km/h and 15 km/h. Find boat speed and stream speed.
A boat's speed in still water is 24 km/h and stream speed is 6 km/h. Find the time required to cover 90 km downstream.
A boat travels 72 km downstream at 24 km/h and returns upstream at 18 km/h. Find total time and average speed.
Downstream and upstream speeds are in the ratio 5:3. Find the ratio of boat speed to stream speed.

20. ANSWERS

``` ```

Q	Answer	Q	Answer
1	Down = 22 km/h; Up = 14 km/h	6	15 km/h
2	Boat = 18 km/h; Stream = 4 km/h	7	Boat = 20 km/h; Stream = 5 km/h
3	3 hours	8	3 hours
4	4 hours	9	7 hours; Average = 20⅗ km/h
5	5 hours	10	4 : 1

21. EXAM SHORTCUTS

``` Shortcut 1: Downstream means B + S. ```

``` Shortcut 2: Upstream means B − S. ```

``` Shortcut 3: Boat speed = half of the sum of downstream and upstream speeds. ```

``` Shortcut 4: Stream speed = half of the difference between downstream and upstream speeds. ```

``` Shortcut 5: For equal distances, average speed is 2ab/(a+b), not simply (a+b)/2. ```

``` Common Mistake: Never add stream speed when calculating upstream speed. The current opposes the boat, so it must be subtracted. ```

``` UTKALPRESS — ARITHMETIC SERIES

Master Formula:
Downstream = Boat + Stream
Upstream = Boat − Stream
Boat = (Downstream + Upstream)/2
Stream = (Downstream − Upstream)/2 ```