Question 212 marksVery Short AnswerSection B

A man in a boat goes 12 km downstream and comes back to the starting point by rowing non-stop in a total time of 3 hours. If the speed of the stream is 3 km/h, find the speed with which the man can row the boat in still water.

Numbers, Quantification and Numerical Applications
Boats and Streams — Speed in Still Water

Official answer

The man can row at 9 km/h in still water.


Set up: Let the still-water speed be x km/h. With stream 3 km/h:

  • Downstream speed = (x+3)(x + 3), upstream speed = (x−3)(x - 3).

Time equation: 12x+3+12x−3=3\frac{12}{x+3} + \frac{12}{x-3} = 3, which simplifies to x2−8x−9=0x^2 - 8x - 9 = 0, giving x=9x = 9 (rejecting x=−1x = -1).


Answer: Speed in still water = 9 km/h.

Answer keywords:boats and streamsdownstream speedupstream speedstill water speedrelative speedquadratic equationtime distance speed

Marking scheme

  1. 1.0.5 mark: correct expressions for downstream (x+3)(x+3) and upstream (x−3)(x-3) speeds.
  2. 2.1 mark: forming and simplifying the equation 12/(x+3)+12/(x−3)=312/(x+3)+12/(x-3)=3 into x2−8x−9=0x^2-8x-9=0.
  3. 3.0.5 mark: solving to get x=9x = 9 km/h and rejecting x=−1x = -1.

Hint

Downstream speed = x+3x+3, upstream = x−3x-3; add both times, set equal to 3, and solve the quadratic.

Quick oral answer

Taking still-water speed as x, downstream is x plus 3 and upstream is x minus 3; adding the two times of 12 km each to get 3 hours gives x squared minus 8x minus 9 equals zero, so x is 9 km per hour.

Common mistakes

  1. 1.Adding the stream speed to distance or forgetting to use (x+3)(x+3) and (x−3)(x-3) as the effective speeds.
  2. 2.Keeping the negative root x=−1x = -1 as a valid answer instead of rejecting it, since speed must be positive.
  3. 3.Arithmetic slip while cross-multiplying, giving a wrong quadratic like x2−8x+9=0x^2-8x+9=0.

Analysis and explanation

This is a classic boats-and-streams problem combining relative speed with the time = distance/speed relationship.


Concept: When rowing with the current (downstream) the effective speed is (boat+stream)(\text{boat} + \text{stream}); against the current (upstream) it is (boat−stream)(\text{boat} - \text{stream}). The total time for the round trip is the sum of the two individual times.


Method: Because the same 12 km is covered each way at different speeds, we add the two times and equate to 3 hours. Clearing denominators gives a quadratic whose positive root is the physical answer.


Exam trap: A speed cannot be negative, so the root x=−1x = -1 is discarded. Also, x must exceed the stream speed (3 km/h), otherwise upstream motion would be impossible — 9 km/h correctly satisfies x>3x > 3.


Real-world link: The same idea models an aircraft flying with and against a wind, or a swimmer in a river.

Interesting facts

The average speed for the whole trip is not the still-water speed — it is total distance (24 km) over total time (3 h) = 8 km/h, lower than 9 km/h because the slow upstream leg dominates.

The same downstream/upstream model is used in aviation as 'ground speed=airspeed±wind speed\text{ground speed} = \text{airspeed} \pm \text{wind speed}', a core idea in flight-time planning.

Question details

Q21 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section B
Marks
2
Type
Very Short Answer
Chapter
Numbers, Quantification and Numerical Applications
Topic
Boats and Streams — Speed in Still Water
Bloom's level
apply
Syllabus topic
Unit: Numbers, Quantification and Numerical Applications — Boats and Streams
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why do we reject the root x=−1x = -1?

Speed is a physical quantity that cannot be negative, so x=−1x = -1 km/h is meaningless. Only the positive root x=9x = 9 km/h is accepted. Additionally, the still-water speed must exceed the stream speed of 3 km/h for upstream travel to be possible, which 9 km/h satisfies.

How is downstream and upstream speed defined?

Downstream speed is the boat's still-water speed plus the stream speed, because the current helps the boat. Upstream speed is the still-water speed minus the stream speed, because the boat rows against the current. Here they are (x+3)(x+3) and (x−3)(x-3) km/h respectively.