Episode 8 — Aptitude and Reasoning / 8.13 — Boats and Streams
8.13.c Solved Examples -- Boats and Streams
Problem 1: Finding Downstream and Upstream Speed (Easy)
A man can row at 8 km/h in still water. The river flows at 2 km/h. Find his downstream and upstream speeds.
B = 8 km/h, S = 2 km/h
Downstream speed = B + S = 8 + 2 = 10 km/h
Upstream speed = B - S = 8 - 2 = 6 km/h
Answer: Downstream = 10 km/h, Upstream = 6 km/h
Problem 2: Finding Boat and Stream Speed (Easy)
A boat goes 14 km/h downstream and 8 km/h upstream. Find the speed of the boat in still water and the speed of the stream.
Downstream speed (D_s) = 14 km/h
Upstream speed (U_s) = 8 km/h
B = (D_s + U_s) / 2 = (14 + 8) / 2 = 22/2 = 11 km/h
S = (D_s - U_s) / 2 = (14 - 8) / 2 = 6/2 = 3 km/h
Answer: Boat speed = 11 km/h, Stream speed = 3 km/h
Problem 3: Time to Travel Downstream (Easy)
A boat with speed 12 km/h in still water goes downstream in a river flowing at 3 km/h. How long to cover 45 km?
Downstream speed = 12 + 3 = 15 km/h
Time = 45 / 15 = 3 hours
Answer: 3 hours
Problem 4: Time to Travel Upstream (Easy)
Using the same boat and river from Problem 3, how long to cover 45 km upstream?
Upstream speed = 12 - 3 = 9 km/h
Time = 45 / 9 = 5 hours
Answer: 5 hours
Problem 5: Finding Distance from Times (Easy)
A man rows downstream for 2 hours and upstream for 3 hours, covering the same distance. His still-water speed is 5 km/h. Find the stream speed and distance.
Distance downstream = Distance upstream
(B + S) x 2 = (B - S) x 3
(5 + S) x 2 = (5 - S) x 3
10 + 2S = 15 - 3S
5S = 5
S = 1 km/h
Distance = (5 + 1) x 2 = 12 km
Verification: Upstream = (5 - 1) x 3 = 12 km (correct)
Answer: Stream speed = 1 km/h, Distance = 12 km
Problem 6: Round Trip Time (Moderate)
A boat can travel at 10 km/h in still water. The stream speed is 2 km/h. Find the time for a round trip of 48 km (24 km each way).
Downstream speed = 10 + 2 = 12 km/h
Upstream speed = 10 - 2 = 8 km/h
Time downstream = 24 / 12 = 2 hours
Time upstream = 24 / 8 = 3 hours
Total time = 2 + 3 = 5 hours
Answer: 5 hours
Problem 7: Average Speed for Round Trip (Moderate)
From Problem 6, find the average speed for the round trip.
Total distance = 24 + 24 = 48 km
Total time = 5 hours
Average speed = 48 / 5 = 9.6 km/h
Using formula: (B^2 - S^2)/B = (100 - 4)/10 = 96/10 = 9.6 km/h (confirmed)
Alternative: 2 x 12 x 8 / (12 + 8) = 192/20 = 9.6 km/h
Answer: 9.6 km/h
Note: This is LESS than the still water speed of 10 km/h.
Problem 8: Finding Stream Speed from Times (Moderate)
A boat takes 3 hours to go 36 km downstream and 6 hours to return. Find the speed of the stream.
Downstream speed = 36 / 3 = 12 km/h = B + S
Upstream speed = 36 / 6 = 6 km/h = B - S
S = (12 - 6) / 2 = 6 / 2 = 3 km/h
B = (12 + 6) / 2 = 18 / 2 = 9 km/h
Answer: Stream speed = 3 km/h
Problem 9: Finding Distance from Round Trip (Moderate)
A man rows at 6 km/h in still water. The stream flows at 2 km/h. If a round trip takes 12 hours, find the one-way distance.
Downstream speed = 6 + 2 = 8 km/h
Upstream speed = 6 - 2 = 4 km/h
Let one-way distance = D
D/8 + D/4 = 12
D/8 + 2D/8 = 12
3D/8 = 12
D = 32 km
Verification:
T_down = 32/8 = 4 hours
T_up = 32/4 = 8 hours
Total = 12 hours (correct)
Answer: 32 km
Problem 10: The "n Times Longer" Problem (Moderate)
A boat takes twice as long to go upstream as downstream for the same distance. If the stream speed is 3 km/h, find the speed of the boat in still water.
T_up = 2 x T_down
Using the formula: B/S = (n+1)/(n-1) where n = 2
B/3 = (2+1)/(2-1) = 3/1
B = 9 km/h
Verification:
Downstream speed = 9 + 3 = 12 km/h
Upstream speed = 9 - 3 = 6 km/h
Ratio of times = 12/6 = 2 (upstream takes 2x, correct)
Answer: 9 km/h
Problem 11: Boat and Current Speed from Two Distances (Moderate)
A boat covers 24 km upstream and 36 km downstream in 6 hours each. Find the speed of the boat and the current.
Wait -- 6 hours each means:
Upstream speed = 24/6 = 4 km/h = B - S
Downstream speed = 36/6 = 6 km/h = B + S
B = (6 + 4)/2 = 5 km/h
S = (6 - 4)/2 = 1 km/h
Answer: Boat = 5 km/h, Current = 1 km/h
Problem 12: Meeting Problem on a River (Moderate)
Two boats start simultaneously from points A and B, 60 km apart on a river. Boat 1 goes downstream from A at 8 km/h (still water), Boat 2 goes upstream from B at 12 km/h (still water). Stream speed is 2 km/h. When do they meet?
A ========== 60 km ===========> B
Boat 1 ----> (downstream) <---- Boat 2 (upstream)
current ---->
Key insight: Stream cancels in relative speed!
Relative speed = 8 + 12 = 20 km/h (sum of still-water speeds)
Time to meet = 60 / 20 = 3 hours
Location from A:
Boat 1 effective speed = 8 + 2 = 10 km/h
Distance from A = 10 x 3 = 30 km (meeting point)
Answer: 3 hours, 30 km from A
Problem 13: Floating Object (The Hat Problem) (Advanced)
A man rows upstream. After 3 km, his hat falls into the water. He continues upstream for 30 minutes, then turns back. He catches the hat 2 km downstream from where it fell. Find the stream speed.
Let B = boat speed, S = stream speed.
Hat falls at point P and floats downstream.
After 30 min, man turns back.
Man catches hat at point Q, which is 2 km downstream from P.
Time for hat to float from P to Q = 2/S hours.
During the same total time (from when hat fell to when caught):
- Man went upstream for 30 min = 0.5 hours
- Man turned back and rowed downstream for some time t.
- Total time since hat fell = 0.5 + t hours
Using the frame-of-reference trick:
In the water's frame, the hat is stationary.
Man rows away at speed B for 0.5 hours (distance = 0.5B).
Man rows back at speed B for 0.5 hours to return to hat.
So time after turning back = 0.5 hours.
Total time since hat fell = 0.5 + 0.5 = 1 hour.
In this 1 hour, the hat floated 2 km.
Stream speed = 2/1 = 2 km/h.
Answer: 2 km/h
Problem 14: Finding Distance When Total Time is Given (Advanced)
A person rows 10 km/h in still water. Stream is 5 km/h. He rows to a place and comes back in 15 hours. Find the distance.
Downstream speed = 10 + 5 = 15 km/h
Upstream speed = 10 - 5 = 5 km/h
D/15 + D/5 = 15
D/15 + 3D/15 = 15
4D/15 = 15
D = 15 x 15/4 = 225/4 = 56.25 km
Answer: 56.25 km
Problem 15: Three Downstream Trips (Advanced)
A boat goes 36 km downstream in 4 hours and 48 km upstream in 8 hours. How long will it take to cover 18 km in still water?
Downstream speed = 36/4 = 9 km/h = B + S
Upstream speed = 48/8 = 6 km/h = B - S
B = (9 + 6)/2 = 7.5 km/h
S = (9 - 6)/2 = 1.5 km/h
In still water, speed = B = 7.5 km/h
Time for 18 km = 18/7.5 = 2.4 hours = 2 hours 24 minutes
Answer: 2 hours 24 minutes
Problem 16: Speed Ratio Problem (Advanced)
The speed of a boat in still water is to the speed of the stream as 5:1. The boat travels 36 km upstream. Find the time if the downstream journey of the same distance takes 4 hours.
B : S = 5 : 1. Let B = 5k, S = k.
Downstream speed = 5k + k = 6k
Upstream speed = 5k - k = 4k
Downstream time for 36 km = 4 hours
36 / 6k = 4
6k = 9
k = 1.5
B = 7.5 km/h, S = 1.5 km/h
Upstream speed = 4k = 6 km/h
Time upstream = 36 / 6 = 6 hours
Answer: 6 hours
Problem 17: Two People in Different Boats (Advanced)
A and B start rowing from the same point. A goes downstream and B goes upstream. After 1 hour, they are 16 km apart. If the stream speed is 2 km/h and A's still-water speed is 8 km/h, find B's still-water speed.
A goes downstream: distance = (8 + 2) x 1 = 10 km (from start, downstream)
B goes upstream: distance = (Bb - 2) x 1 = (Bb - 2) km (from start, upstream)
They are going in OPPOSITE directions (one downstream, one upstream).
Distance apart = 10 + (Bb - 2) = 16
Bb - 2 = 6
Bb = 8 km/h
But wait -- stream should cancel:
Relative speed = 8 + Bb (sum of still-water speeds, since stream cancels)
Distance in 1 hour = 8 + Bb = 16
Bb = 8 km/h
Both methods confirm Bb = 8 km/h.
Actually let me recheck the first method:
10 + (8-2) = 10 + 6 = 16. Correct.
Answer: B's still-water speed = 8 km/h
Problem 18: Upstream Time with Increased Current (Advanced)
A man rows 40 km upstream in 8 hours when the stream speed is 2 km/h. If the stream speed increases to 4 km/h, how long will the upstream trip take?
Original: Upstream speed = 40/8 = 5 km/h = B - 2
So B = 7 km/h
New upstream speed = B - 4 = 7 - 4 = 3 km/h
New time = 40/3 = 13.33 hours = 13 hours 20 minutes
Answer: 13 hours 20 minutes
Problem 19: Ratio of Times and Finding Stream Speed (Advanced)
A boat covers a distance downstream in 4 hours and the same distance upstream in 6 hours. If the still-water speed is 10 km/h, find the stream speed and the distance.
D = (B + S) x 4 = (B - S) x 6
(10 + S) x 4 = (10 - S) x 6
40 + 4S = 60 - 6S
10S = 20
S = 2 km/h
Distance = (10 + 2) x 4 = 48 km
Verification: (10 - 2) x 6 = 48 km (correct)
Answer: Stream speed = 2 km/h, Distance = 48 km
Problem 20: Escalator Analogy Problem (Advanced)
A man walks up a moving escalator. Walking at his normal speed, he takes 30 steps to reach the top. If he walks twice as fast, he takes 40 steps. How many steps are visible on the escalator?
Let man's normal speed = v steps/sec
Escalator speed = e steps/sec
Total visible steps = N
Case 1: Normal speed
Man takes 30 steps. Time = 30/v seconds.
Escalator moves 30e/v steps in this time.
N = 30 + 30e/v = 30(1 + e/v) ...(1)
Case 2: Double speed (2v)
Man takes 40 steps. Time = 40/(2v) = 20/v seconds.
Escalator moves 20e/v steps.
N = 40 + 20e/v ...(2)
From (1): N = 30 + 30e/v --> 30e/v = N - 30
From (2): N = 40 + 20e/v --> 20e/v = N - 40
Dividing: (30e/v)/(20e/v) = (N-30)/(N-40)
3/2 = (N-30)/(N-40)
3(N-40) = 2(N-30)
3N - 120 = 2N - 60
N = 60
Answer: 60 steps visible on the escalator
Problem 21: Boat Race on a River (Advanced)
In a boat race, boat A has a still-water speed of 12 km/h and boat B has 10 km/h. The race is 6 km downstream followed by 6 km upstream. Stream speed is 2 km/h. By how much time does A win?
Boat A:
Downstream: 6/(12+2) = 6/14 = 3/7 hours
Upstream: 6/(12-2) = 6/10 = 3/5 hours
Total A = 3/7 + 3/5 = (15 + 21)/35 = 36/35 hours
Boat B:
Downstream: 6/(10+2) = 6/12 = 1/2 hours
Upstream: 6/(10-2) = 6/8 = 3/4 hours
Total B = 1/2 + 3/4 = 5/4 hours
Difference = 5/4 - 36/35 = (175 - 144)/140 = 31/140 hours
= 31/140 x 60 = 1860/140 = 13.29 minutes
Answer: A wins by approximately 13.3 minutes (31/140 hours)
Problem 22: Current Speed Changes Mid-Journey (Advanced)
A boat rows 20 km upstream in 4 hours. For the first 10 km, the stream speed is 1 km/h, and for the next 10 km, the stream speed is 3 km/h. Find the boat's still-water speed.
Let B = still-water speed.
Time for first 10 km upstream = 10/(B - 1)
Time for next 10 km upstream = 10/(B - 3)
Total: 10/(B-1) + 10/(B-3) = 4
Let us try B = 6:
10/5 + 10/3 = 2 + 3.33 = 5.33 (too much)
Try B = 8:
10/7 + 10/5 = 1.43 + 2 = 3.43 (too little)
Try B = 7:
10/6 + 10/4 = 1.67 + 2.5 = 4.17 (close but slightly more)
Try B = 7.5:
10/6.5 + 10/4.5 = 1.538 + 2.222 = 3.76 (less)
Solving algebraically:
10(B-3) + 10(B-1) = 4(B-1)(B-3)
10B - 30 + 10B - 10 = 4(B^2 - 4B + 3)
20B - 40 = 4B^2 - 16B + 12
4B^2 - 36B + 52 = 0
B^2 - 9B + 13 = 0
B = (9 + sqrt(81-52))/2 = (9 + sqrt(29))/2 = (9 + 5.385)/2 = 7.19 km/h
Answer: Approximately 7.19 km/h
Problem 23: Relative Speed on River (Advanced)
Two boats start 80 km apart on a river. Boat A (still water: 15 km/h) goes downstream while Boat B (still water: 25 km/h) goes upstream towards A. Stream speed is 5 km/h. When and where do they meet?
Boat A downstream speed = 15 + 5 = 20 km/h
Boat B upstream speed = 25 - 5 = 20 km/h
They approach each other:
Relative speed = 20 + 20 = 40 km/h
(Or using shortcut: 15 + 25 = 40 km/h, stream cancels)
Time = 80/40 = 2 hours
Distance from A's start = 20 x 2 = 40 km
Distance from B's start = 20 x 2 = 40 km
They meet exactly in the middle!
Answer: 2 hours, 40 km from each starting point
Problem 24: Swimming Problem (Advanced)
A swimmer can swim at 6 km/h in still water. The river is 1 km wide and flows at 2 km/h. If the swimmer aims straight across, how far downstream is he carried?
Width of river = 1 km
Swimming speed (perpendicular to bank) = 6 km/h
Current speed (along the bank) = 2 km/h
Time to cross = 1/6 hours (only the perpendicular component matters)
Downstream drift = current x time = 2 x (1/6) = 1/3 km = 333.33 m
Answer: 333.33 m downstream
Problem 25: Complex Round Trip with Wind (Advanced)
An aircraft flies 1200 km with a tailwind at an effective speed of 300 km/h, then returns against the wind at 200 km/h. What is the wind speed and the aircraft's speed in still air?
This is the same as boats and streams:
Tailwind = downstream, Headwind = upstream
B + S = 300 (with tailwind)
B - S = 200 (against wind)
B = (300 + 200)/2 = 250 km/h (aircraft speed in still air)
S = (300 - 200)/2 = 50 km/h (wind speed)
Round trip time = 1200/300 + 1200/200 = 4 + 6 = 10 hours
Average speed = 2400/10 = 240 km/h
Using formula: 2 x 300 x 200 / (300+200) = 120000/500 = 240 (confirmed)
Answer: Aircraft = 250 km/h, Wind = 50 km/h
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