Boats and Streams – Complete Short Notes for Competitive Exams

1. Basic Concept

Boats and streams problems are based on Relative Speed (concept of motion in still water vs. flowing water).

Water current affects the speed of the boat depending on direction of motion:

Downstream (with current): Boat moves faster
Upstream (against current): Boat moves slower

2. Key Terms

Term	Meaning
Speed of boat in still water	$b$ km/hr
Speed of stream (current)	$s$ km/hr
Downstream speed	$b$
Upstream speed	$b−$

3. Formulas

Concept	Formula
Downstream speed	$b + s$
Upstream speed	$b - s$
Speed of boat (still water)	$\frac{(Down + Up)}{2}$
Speed of stream	$\frac{(Down - Up)}{2}$
Time =	Distance / Speed

Example:

Downstream speed = 12 km/hr, Upstream speed = 8 km/hr
Then:

Speed of boat = (12 + 8)/2 = 10 km/hr
Speed of stream = (12 − 8)/2 = 2 km/hr

4. Relationship Between Distance, Speed, and Time

Speed = \frac{Distance}{Time}, Time = \frac{Distance}{Speed​}

Always use same unit (km/hr or m/s).

5. Unit Conversion

1 \text{ km/hr} = \frac{5}{18} \text{ m/s}

1 \text{ m/s} = \frac{18}{5} \text{ km/hr}

6. Type 1 – Find Boat or Stream Speed

If Downstream (D) and Upstream (U) speeds are known:

b = \frac{D + U}{2}, s = \frac{D - U}{2​}

Example:

A boat goes 16 km downstream in 2 hrs and returns in 4 hrs.
Downstream speed = 8 km/hr, Upstream speed = 4 km/hr
So, $b = 6$ , $s = 2$ .

7. Type 2 – Find Time

If distance = d, downstream speed = (b + s), upstream speed = (b − s):

Time downstream = $\frac{d}{b + s}$
Time upstream = $\frac{d}{b - s}$

Total time = sum of both.

Example:

Boat speed = 10 km/hr, Stream speed = 2 km/hr, Distance = 24 km
Downstream time = 24/12 = 2 hr
Upstream time = 24/8 = 3 hr
Total = 5 hr

8. Type 3 – Average Speed (Round Trip)

Average Speed = \frac{2 \times (D o w n s t r e a m \times U p s t r e a m)}{(D o w n s t r e a m + U p s t r e a m)​}

Example:

Downstream = 12 km/hr, Upstream = 8 km/hr
Avg speed = (2×12×8)/(12+8) = 192/20 = 9.6 km/hr

9. Type 4 – Time Difference Question

If boat travels same distance downstream and upstream,

\text{Time difference} = d \times \left(\frac{1}{b - s} - \frac{1}{b + s}\right)

Example:

Boat = 12 km/hr, Stream = 3 km/hr, Distance = 30 km
ΔT = 30 × (1/9 − 1/15) = 30 × (6/135 − 4/135) = 1/1.5 = 2 hr

10. Type 5 – When One Direction Time Known

If boat takes t₁ hrs downstream and t₂ hrs upstream for same distance:

b = \frac{d}{2} \left(\frac{1}{t₁} + \frac{1}{t₂}\right)

s = \frac{d}{2} \left(\frac{1}{t₁} - \frac{1}{t₂}\right)

Example:

Distance = 20 km, Downstream = 2 hr, Upstream = 4 hr
b = ½(10 + 5) = 7.5 km/hr, s = ½(10 − 5) = 2.5 km/hr

11. Type 6 – Boat in Still Water vs Stream

If a boat covers d km in still water in t₁ hr and same distance in downstream in t₂ hr,
then current speed can be derived from the difference.

Example:

Boat covers 24 km in still water in 3 hr (speed = 8 km/hr).
Downstream takes 2 hr → stream = D − b = 12 − 8 = 4 km/hr

12. Type 7 – Relative Speed Concept

When two boats move in opposite directions:

Relative Speed = Sum of speeds

When two boats move in same direction:

Relative Speed = Difference of speeds

13. Type 8 – Man Swimming in River

If man swims at m km/hr, stream speed = s km/hr:

Case	Effective Speed
With current	m + s
Against current	m − s

Example:

Swimmer = 5 km/hr, Stream = 1 km/hr
Upstream = 4 km/hr, Downstream = 6 km/hr

If he swims 12 km downstream → time = 12/6 = 2 hr

14. Type 9 – Average Velocity Concept

\text{Average speed for equal distance} = \frac{2ab}{a + b}

Same as time & work / relative motion problems.

15. Conversion Shortcuts

| km/hr → m/s | × 5/18 |
| m/s → km/hr | × 18/5 |
| 1 hr | 3600 s |
| 1 km | 1000 m |

16. Practice Questions

Q1. Boat speed = 12 km/hr, Stream = 3 km/hr. Find downstream & upstream speeds.
→ Downstream = 15, Upstream = 9.

Q2. Distance = 36 km, Downstream = 15, Upstream = 9 → total time = 36/15 + 36/9 = 2.4 + 4 = 6.4 hr

Q3. Downstream 18, Upstream 12 → Find boat and stream.
b = (18 + 12)/2 = 15, s = (18 − 12)/2 = 3 km/hr.

17. Quick Trick Table

Quantity	Formula
Downstream speed	b + s
Upstream speed	b − s
Boat speed	(Down + Up)/2
Stream speed	(Down − Up)/2
Average speed (round trip)	2DU/(D + U)
Time difference	d[(1/(b−s)) − (1/(b+s))]

18. Common Errors to Avoid

❌ Forgetting that downstream = b + s and upstream = b − s
❌ Mixing distance/time ratios
❌ Not keeping consistent units
❌ Ignoring equal distance condition for average speed

19. Real-life Applications

Riverboat travel timing
Current flow speed estimation
Swimming competitions in river currents
Ship navigation & fuel efficiency planning

20. Quick Revision Chart

Concept	Formula	Note
Downstream	b + s	With flow
Upstream	b − s	Against flow
Boat speed	(D + U)/2	Average
Stream speed	(D − U)/2	Difference
Average speed	2DU/(D + U)	Round trip
Time diff	d[(1/(b−s)) − (1/(b+s))]	Same distance