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.

Q: 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.

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 = $(x + 3)$, upstream speed = $(x - 3)$. **Time equation:** $\frac{12}{x+3} + \frac{12}{x-3} = 3$, which simplifies to $x^2 - 8x - 9 = 0$, giving $x = 9$ (rejecting $x = -1$). **Answer:** Speed in still water = **9 km/h**.

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 = $(x + 3)$ , upstream speed = $(x - 3)$ .

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

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

boats and streamsdownstream speedupstream speedstill water speedrelative speedquadratic equationtime distance speed

Marking Scheme

10.5 mark: correct expressions for downstream $(x+3)$ and upstream $(x-3)$ speeds.
21 mark: forming and simplifying the equation $12/(x+3)+12/(x-3)=3$ into $x^2-8x-9=0$ .
30.5 mark: solving to get $x = 9$ km/h and rejecting $x = -1$ .

Hint

Downstream speed = $x+3$ , upstream = $x-3$ ; 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.

Analysis & 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 $(\text{boat} + \text{stream})$ ; against the current (upstream) it is $(\text{boat} - \text{stream})$ . 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 $x = -1$ is discarded. Also, x must exceed the stream speed (3 km/h), otherwise upstream motion would be impossible — 9 km/h correctly satisfies $x > 3$ .

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

Common Mistakes

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

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 ' $\text{ground speed} = \text{airspeed} \pm \text{wind speed}$ ', a core idea in flight-time planning.

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Frequently Asked Questions

Why do we reject the root $x = -1$ ?

Speed is a physical quantity that cannot be negative, so $x = -1$ km/h is meaningless. Only the positive root $x = 9$ 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 $(x+3)$ and $(x-3)$ km/h respectively.