Boats and Streams questions with answers and concepts

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Boats and Streams problems deal with the motion of a boat in still water and in a flowing stream. These problems involve understanding the effect of the stream's current on the boat's effective speed when moving upstream (against the current) or downstream (with the current).

Key formulas and rules

7 rules

Still water speed: $u = \frac{v _{d} + v _{u}}{2}$ (where $v_{d} = downstream speed$ , $v_{u} = upstream speed$ )

Stream speed: $v = \frac{v _{d} - v _{u}}{2}$

Downstream speed: $v_{d} = u + v$

Upstream speed: $v_{u} = u - v$ (condition: $u > v$ for forward motion)

Round-trip average speed: $Avg Speed = \frac{2 v _{d} v _{u}}{v _{d} + v _{u}} = \frac{u ^{2} - v ^{2}}{u}$

Time taken downstream: $t_{d} = \frac{D}{u + v}$ ; Time taken upstream: $t_{u} = \frac{D}{u - v}$

Distance formula from total time $T$ : $D = \frac{T ( u ^{2} - v ^{2} )}{2 u}$

Key concepts

Downstream Motion

When a boat moves downstream, it travels in the same direction as the stream flow. The stream assists the boat, resulting in a higher effective speed. Downstream speed = Speed of boat in still water + Speed of stream. For example, if a boat's speed in still water is 15 km/h and the stream flows at 3 km/h, the downstream speed is 18 km/h.

Upstream Motion

When a boat moves upstream, it travels against the stream flow. The stream opposes the boat, resulting in a lower effective speed. Upstream speed = Speed of boat in still water - Speed of stream. For example, if a boat's speed in still water is 15 km/h and the stream flows at 3 km/h, the upstream speed is 12 km/h. Note: The boat can only move upstream if its speed in still water exceeds the stream speed.

Still Water and Stream Speed

The speed of a boat in still water is its inherent speed without any current assistance or opposition. The speed of the stream (or current) represents how fast the water is flowing. These two speeds combine to give effective speeds for upstream and downstream travel. When solving problems, always identify what is given (still water speed, stream speed, or effective speeds) and what needs to be found.

Round Trip Problems

In round trip problems, a boat travels from point A to point B (either upstream or downstream) and returns. The total time is the sum of times for each leg. Average speed for the entire journey is NOT the arithmetic mean but the harmonic mean: (2 x downstream speed x upstream speed) / (downstream speed + upstream speed). This accounts for the fact that more time is spent traveling at the slower speed.

Floating Objects and Relative Motion

When an object floats with the stream (no self-propulsion), it moves at the speed of the stream. The time taken for a boat to travel downstream and return upstream to catch a floating object dropped at the start is independent of the stream speed—it depends only on the boat's speed in still water. This is because both the boat and the object are affected equally by the stream's motion.

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Worked examples

Example 1

A boat can travel at 18 km/h in still water. If the river current flows at 6 km/h, find: (a) downstream speed, (b) upstream speed, and (c) total time taken to travel 48 km downstream and return.

Solution

Step 1: Calculate downstream and upstream velocities.

v_{d} = u + v = 18 + 6 = 24 km/h

v_{u} = u - v = 18 - 6 = 12 km/h

Step 2: Compute travel times.

t_{down} = \frac{48}{24} = 2 hours

t_{up} = \frac{48}{12} = 4 hours

Step 3: Total round-trip time.

T = t_{down} + t_{up} = 2 + 4 = 6 hours

Answer: (a) 24 km/h, (b) 12 km/h, and (c) 6 hours.

Example 2

A motorboat covers a certain distance downstream at 15 km/h and upstream at 9 km/h. Find the speed of the boat in still water and the speed of the river current.

Solution

Step 1: Calculate speed in still water ( $u$ ).

u = \frac{v _{d} + v _{u}}{2} = \frac{15 + 9}{2} = \frac{24}{2} = 12 km/h

Step 2: Calculate river current velocity ( $v$ ).

v = \frac{v _{d} - v _{u}}{2} = \frac{15 - 9}{2} = \frac{6}{2} = 3 km/h

Answer: Still water speed = 12 km/h; river current = 3 km/h.

Example 3

A rower finds that it takes twice as long to row upstream as to row downstream across the same stretch. If the river flows at 4 km/h, what is the rowing speed in still water?

Solution

Step 1: Relate travel times for identical distance.

t_{u} = 2 t_{d} ⟹ \frac{D}{u - v} = 2 \times \frac{D}{u + v}

u + v = 2 (u - v) ⟹ u + v = 2 u - 2 v ⟹ u = 3 v

Step 2: Substitute stream speed $v = 4 km/h$ .

u = 3 \times 4 = 12 km/h

Answer: 12 km/h.

Example 4

A boat's speed in still water is 10 km/h and the river velocity is 2 km/h. The round trip takes 5 hours. What is the one-way distance travelled?

Solution

Step 1: Apply the round-trip distance formula.

D = \frac{T ( u ^{2} - v ^{2} )}{2 u}

Step 2: Substitute $T = 5$ , $u = 10$ , and $v = 2$ .

D = \frac{5 \times ( 1 0 ^{2} - 2 ^{2} )}{2 \times 10} = \frac{5 \times ( 100 - 4 )}{20} = \frac{5 \times 96}{20} = \frac{96}{4} = 24 km

Answer: 24 km.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

A boat travels 36 km upstream in 6 hours and 48 km downstream in 4 hours. Find the speed of the boat in still water.

8 km/hr
9 km/hr
10 km/hr
11 km/hr

Answer: B. 9 km/hr

Explanation

Upstream speed = 36/6 = 6 km/hr
Downstream speed = 48/4 = 12 km/hr
Boat speed = (Downstream + Upstream)/2 = (12+6)/2 = 9 km/hr

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A swimmer can swim 20 km downstream in 2 hours. If the speed of the stream is 3 km/hr, how long will he take to swim 14 km upstream?

2 hours
2.5 hours
3 hours
3.5 hours

Answer: D. 3.5 hours

Explanation

Downstream speed = 20/2 = 10 km/hr
Swimmer speed in still water = 10 - 3 = 7 km/hr
Upstream speed = 7 - 3 = 4 km/hr
Time for 14 km upstream = 14/4 = 3.5 hours
Answer: 3.5 hours.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The speed of a stream is 3 km/hr. A boat goes 42 km downstream in 3 hours. How long will it take to go 28 km upstream?

3 hours
3.5 hours
4 hours
5 hours

Answer: B. 3.5 hours

Explanation

Downstream speed = 42/3 = 14 km/hr
Boat speed = 14 - 3 = 11 km/hr
Upstream speed = 11 - 3 = 8 km/hr
Time for 28 km upstream = 28/8 = 3.5 hours
Answer: 3.5 hours.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always draw a diagram showing the direction of stream flow and boat movement to visualize upstream vs downstream clearly.
Remember that upstream speed is always less than downstream speed. If your calculation shows otherwise, recheck your work.
For round trip average speed, use the harmonic mean formula: 2ab/(a+b), not the arithmetic mean (a+b)/2.
When given the ratio of times for equal distances upstream and downstream, set up the equation: (still water speed + stream speed) / (still water speed - stream speed) = ratio of times.
In problems where a boat travels a certain distance downstream and returns, remember that upstream travel always takes more time than downstream travel for the same distance.
For floating object problems, the key insight is that both the object and the boat are affected equally by the stream, so relative motion depends only on the boat's still water speed.

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Key concepts

Downstream Motion

Upstream Motion

Still Water and Stream Speed

Round Trip Problems

Floating Objects and Relative Motion

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Worked examples

Example 1

A boat can travel at 18 km/h in still water. If the river current flows at 6 km/h, find: (a) downstream speed, (b) upstream speed, and (c) total time taken to travel 48 km downstream and return.

Solution

Step 1: Calculate downstream and upstream velocities.

v_{d} = u + v = 18 + 6 = 24 km/h

v_{u} = u - v = 18 - 6 = 12 km/h

Step 2: Compute travel times.

t_{down} = \frac{48}{24} = 2 hours

t_{up} = \frac{48}{12} = 4 hours

Step 3: Total round-trip time.

T = t_{down} + t_{up} = 2 + 4 = 6 hours

Answer: (a) 24 km/h, (b) 12 km/h, and (c) 6 hours.

Example 2

A motorboat covers a certain distance downstream at 15 km/h and upstream at 9 km/h. Find the speed of the boat in still water and the speed of the river current.

Solution

Step 1: Calculate speed in still water ( $u$ ).

u = \frac{v _{d} + v _{u}}{2} = \frac{15 + 9}{2} = \frac{24}{2} = 12 km/h

Step 2: Calculate river current velocity ( $v$ ).

v = \frac{v _{d} - v _{u}}{2} = \frac{15 - 9}{2} = \frac{6}{2} = 3 km/h

Answer: Still water speed = 12 km/h; river current = 3 km/h.

Example 3

A rower finds that it takes twice as long to row upstream as to row downstream across the same stretch. If the river flows at 4 km/h, what is the rowing speed in still water?

Solution

Step 1: Relate travel times for identical distance.

t_{u} = 2 t_{d} ⟹ \frac{D}{u - v} = 2 \times \frac{D}{u + v}

u + v = 2 (u - v) ⟹ u + v = 2 u - 2 v ⟹ u = 3 v

Step 2: Substitute stream speed $v = 4 km/h$ .

u = 3 \times 4 = 12 km/h

Answer: 12 km/h.

Example 4

A boat's speed in still water is 10 km/h and the river velocity is 2 km/h. The round trip takes 5 hours. What is the one-way distance travelled?

Solution

Step 1: Apply the round-trip distance formula.

D = \frac{T ( u ^{2} - v ^{2} )}{2 u}

Step 2: Substitute $T = 5$ , $u = 10$ , and $v = 2$ .

D = \frac{5 \times ( 1 0 ^{2} - 2 ^{2} )}{2 \times 10} = \frac{5 \times ( 100 - 4 )}{20} = \frac{5 \times 96}{20} = \frac{96}{4} = 24 km

Answer: 24 km.

Representative solved questions

See the kind of question in this topic before opening the full practice set.

Question 1

A boat travels 36 km upstream in 6 hours and 48 km downstream in 4 hours. Find the speed of the boat in still water.

8 km/hr
9 km/hr
10 km/hr
11 km/hr

Answer: B. 9 km/hr

Explanation

Upstream speed = 36/6 = 6 km/hr
Downstream speed = 48/4 = 12 km/hr
Boat speed = (Downstream + Upstream)/2 = (12+6)/2 = 9 km/hr

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 2

A swimmer can swim 20 km downstream in 2 hours. If the speed of the stream is 3 km/hr, how long will he take to swim 14 km upstream?

2 hours
2.5 hours
3 hours
3.5 hours

Answer: D. 3.5 hours

Explanation

Downstream speed = 20/2 = 10 km/hr
Swimmer speed in still water = 10 - 3 = 7 km/hr
Upstream speed = 7 - 3 = 4 km/hr
Time for 14 km upstream = 14/4 = 3.5 hours
Answer: 3.5 hours.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Question 3

The speed of a stream is 3 km/hr. A boat goes 42 km downstream in 3 hours. How long will it take to go 28 km upstream?

3 hours
3.5 hours
4 hours
5 hours

Answer: B. 3.5 hours

Explanation

Downstream speed = 42/3 = 14 km/hr
Boat speed = 14 - 3 = 11 km/hr
Upstream speed = 11 - 3 = 8 km/hr
Time for 28 km upstream = 28/8 = 3.5 hours
Answer: 3.5 hours.

Sources and review notes

This is an AISEA-authored practice question.

Review status: accepted · Reviewed 2026-08-13 · structure and answer-key checks, editorial quality checks, duplicate screening

Common mistakes and useful habits

Always draw a diagram showing the direction of stream flow and boat movement to visualize upstream vs downstream clearly.
Remember that upstream speed is always less than downstream speed. If your calculation shows otherwise, recheck your work.
For round trip average speed, use the harmonic mean formula: 2ab/(a+b), not the arithmetic mean (a+b)/2.
When given the ratio of times for equal distances upstream and downstream, set up the equation: (still water speed + stream speed) / (still water speed - stream speed) = ratio of times.
In problems where a boat travels a certain distance downstream and returns, remember that upstream travel always takes more time than downstream travel for the same distance.
For floating object problems, the key insight is that both the object and the boat are affected equally by the stream, so relative motion depends only on the boat's still water speed.

Key formulas and rules

Key concepts

Downstream Motion

Upstream Motion

Still Water and Stream Speed

Round Trip Problems

Floating Objects and Relative Motion

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.

Key formulas and rules

Key concepts

Downstream Motion

Upstream Motion

Still Water and Stream Speed

Round Trip Problems

Floating Objects and Relative Motion

Worked examples

Example 1

Solution

Example 2

Solution

Example 3

Solution

Example 4

Solution

Representative solved questions

Question 1

Question 2

Question 3

Common mistakes and useful habits

Ready to test your understanding?

Keep building the same skill family.