Quantitative Techniques · Micro-test
Boats and Streams
Every boats and streams question rests on two sums: the stream adds to the boat's speed going down and takes away from it going up. Fix those two and each question is a line or two of arithmetic.
10 questions · 5 minutes · instant scoring
What this topic actually tests
Let the boat's speed in still water be b and the stream's speed be s, both in km/h. Going downstream the water carries the boat, so its speed is b + s. Going upstream the water works against it, so its speed is b − s. Turned around, if you know the two speeds, the boat's speed in still water is half their sum and the stream's speed is half their difference. Time is distance divided by the speed for that leg, so a round trip is two separate divisions added together, not one division by an average. The average speed for a round trip over the same distance is (b² − s²) ÷ b, which is always less than b: the slow leg lasts longer than the fast one. Worked example: a boat that does 12 km/h in still water on a stream of 4 km/h travels downstream at 16 km/h and upstream at 8 km/h. Thirty-two kilometres each way takes 2 hours down and 4 hours up, 6 hours in all, an average of 64 ÷ 6 = 10 2/3 km/h.
The common trap on this topic
The common mistake is to average the downstream and upstream speeds to get the round-trip speed. That gives the boat's speed in still water, not the average speed of the journey, because the boat spends longer on the slow upstream leg. Work out each leg's time separately and divide total distance by total time. A second slip is mixing up the two halves of the formula: half the sum of the speeds is the boat, half the difference is the stream.
Take the micro-test
A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. What is the boat's speed downstream?
A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. What is the boat's speed upstream?
A boat travels at 20 km/h downstream and 12 km/h upstream. What is its speed in still water?
A boat travels at 20 km/h downstream and 12 km/h upstream. What is the speed of the stream?
A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long does it take to travel 36 km downstream?
A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long does it take to travel 48 km upstream?
A man rows 24 km downstream in 2 hours and the same 24 km upstream in 3 hours. What is his rowing speed in still water?
A boat's speed in still water is 12 km/h and the stream flows at 4 km/h. How long does it take to go 32 km downstream and come back?
A boat's speed in still water is 12 km/h and the stream flows at 4 km/h. What is its average speed for a round trip over the same distance?
A boat takes twice as long to go a distance upstream as to go the same distance downstream. What is the ratio of the boat's speed in still water to the speed of the stream?
Boats and Streams: answers and explanations
A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. What is the boat's speed downstream?
Answer: D. 18 km/h
Downstream the stream adds to the boat's speed: 15 + 3 = 18 km/h.
A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. What is the boat's speed upstream?
Answer: B. 12 km/h
Upstream the stream works against the boat: 15 − 3 = 12 km/h.
A boat travels at 20 km/h downstream and 12 km/h upstream. What is its speed in still water?
Answer: A. 16 km/h
The speed in still water is half the sum of the two speeds: (20 + 12) ÷ 2 = 16 km/h.
A boat travels at 20 km/h downstream and 12 km/h upstream. What is the speed of the stream?
Answer: B. 4 km/h
The stream's speed is half the difference of the two speeds: (20 − 12) ÷ 2 = 4 km/h.
A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long does it take to travel 36 km downstream?
Answer: C. 3 hours
Downstream speed is 10 + 2 = 12 km/h, so 36 km takes 36 ÷ 12 = 3 hours.
A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long does it take to travel 48 km upstream?
Answer: B. 6 hours
Upstream speed is 10 − 2 = 8 km/h, so 48 km takes 48 ÷ 8 = 6 hours.
A man rows 24 km downstream in 2 hours and the same 24 km upstream in 3 hours. What is his rowing speed in still water?
Answer: A. 10 km/h
Downstream speed is 24 ÷ 2 = 12 km/h and upstream speed is 24 ÷ 3 = 8 km/h. Half their sum is (12 + 8) ÷ 2 = 10 km/h.
A boat's speed in still water is 12 km/h and the stream flows at 4 km/h. How long does it take to go 32 km downstream and come back?
Answer: A. 6 hours
Downstream: 32 ÷ 16 = 2 hours. Upstream: 32 ÷ 8 = 4 hours. The round trip takes 6 hours.
A boat's speed in still water is 12 km/h and the stream flows at 4 km/h. What is its average speed for a round trip over the same distance?
Answer: A. 10 2/3 km/h
The average speed is (b² − s²) ÷ b = (144 − 16) ÷ 12 = 128 ÷ 12 = 10 2/3 km/h. Check with 32 km each way: 64 km in 6 hours is 10 2/3 km/h.
A boat takes twice as long to go a distance upstream as to go the same distance downstream. What is the ratio of the boat's speed in still water to the speed of the stream?
Answer: C. 3 : 1
Twice the time means half the speed, so b + s = 2(b − s). That gives b = 3s, a ratio of 3 : 1.
FAQ
What is the formula for downstream and upstream speed?
Downstream speed is the boat's speed in still water plus the stream's speed. Upstream speed is the boat's speed minus the stream's speed.
How do I find the speed of the stream from the two speeds?
Take half the difference of the downstream and upstream speeds. Half their sum is the boat's speed in still water.
Is the average speed of a round trip the boat's speed in still water?
No. The average speed of a round trip is (b² − s²) ÷ b, which is less than b, because the boat spends longer going upstream.
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