Episode 8 — Aptitude and Reasoning / 8.13 — Boats and Streams
8.13.a Concepts and Formulas -- Boats and Streams
1. Core Terminology
Still Water Speed (B): The speed of the boat when there is no current.
Also called "speed of the boat in still water."
Stream Speed (S): The speed of the river current (or stream).
Downstream: Moving WITH the current (in the direction of flow).
Upstream: Moving AGAINST the current (opposite to the flow).
Visual Representation
River flowing from left to right:
~~~~~~~~~~~~ current direction ~~~~~~~~~~~~>
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~>
DOWNSTREAM (with current):
Boat ----> Current ---->
Effective speed = B + S (current helps)
UPSTREAM (against current):
<---- Boat Current ---->
Effective speed = B - S (current resists)
2. Fundamental Formulas
+----------------------------------------------+
| |
| Downstream speed: D_s = B + S |
| Upstream speed: U_s = B - S |
| |
| Where: |
| B = Speed of boat in still water |
| S = Speed of the stream/current |
| |
+----------------------------------------------+
Deriving B and S from Downstream and Upstream Speeds
If downstream speed and upstream speed are known:
Adding: D_s + U_s = (B + S) + (B - S) = 2B
Subtracting: D_s - U_s = (B + S) - (B - S) = 2S
+----------------------------------------------+
| |
| B = (D_s + U_s) / 2 |
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| S = (D_s - U_s) / 2 |
| |
+----------------------------------------------+
Where:
D_s = downstream speed
U_s = upstream speed
3. Time Formulas
Downstream Travel Time
Time_downstream = Distance / (B + S)
Upstream Travel Time
Time_upstream = Distance / (B - S)
Key Insight
For the same distance:
Time_upstream > Time_downstream (ALWAYS)
Because (B - S) < (B + S), so dividing D by a smaller number
gives a larger time.
Ratio: T_up / T_down = (B + S) / (B - S)
4. Round Trip Problems
A boat goes downstream a distance D and then returns upstream the same distance.
A ========= D km ==========> B
downstream --->
<--- upstream
Time downstream = D / (B + S)
Time upstream = D / (B - S)
Total time = D/(B+S) + D/(B-S)
= D[(B-S) + (B+S)] / [(B+S)(B-S)]
= D[2B] / [B^2 - S^2]
+----------------------------------------------+
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| Total round trip time = 2DB / (B^2 - S^2) |
| |
+----------------------------------------------+
Average Speed for Round Trip
Total distance = 2D
Total time = 2DB / (B^2 - S^2)
Average speed = 2D / [2DB / (B^2 - S^2)]
= (B^2 - S^2) / B
+----------------------------------------------+
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| Avg speed (round trip) = (B^2 - S^2) / B |
| |
+----------------------------------------------+
Note: This is ALWAYS less than B (the still water speed).
5. Finding Distance from Round Trip Time
If total round trip time T is given:
T = D/(B+S) + D/(B-S)
T = 2DB / (B^2 - S^2)
D = T(B^2 - S^2) / (2B)
6. Speed of Current from Upstream and Downstream Times
If a boat covers the same distance in T_down downstream and T_up upstream:
D = (B + S) x T_down = (B - S) x T_up
From this:
B + S T_up
----- = ------
B - S T_down
Cross multiply:
(B + S) x T_down = (B - S) x T_up
B.T_down + S.T_down = B.T_up - S.T_up
B(T_down - T_up) = -S(T_down + T_up)
Since T_up > T_down, let us rearrange:
B(T_up - T_down) = S(T_up + T_down)
S/B = (T_up - T_down) / (T_up + T_down)
7. Relative Motion in Streams
Two Boats Moving in the Same Direction
Boat A ----> (speed = B_a) current ---->
Boat B ----> (speed = B_b)
Effective speed of A = B_a + S
Effective speed of B = B_b + S
Relative speed = (B_a + S) - (B_b + S) = B_a - B_b
NOTE: The stream speed cancels out!
The relative speed of two boats moving in the same direction
on the same river is just the difference of their still-water speeds.
Two Boats Moving in Opposite Directions
Boat A ----> (downstream: B_a + S)
<---- Boat B (upstream: B_b - S)
Relative speed of approach = (B_a + S) + (B_b - S) = B_a + B_b
NOTE: Again, stream speed cancels out!
Important Insight
+-------------------------------------------------------+
| |
| When two boats are on the SAME river, the stream |
| speed cancels out in relative speed calculations. |
| |
| This simplifies meeting/overtaking problems greatly. |
| |
+-------------------------------------------------------+
8. Meeting Problems on a River
Two Boats Start from Opposite Ends
A ============ D km ============= B
Boat 1 ----> (current ---->) <---- Boat 2
(downstream) (upstream)
Speed of Boat 1 = B1 + S (downstream)
Speed of Boat 2 = B2 - S (upstream)
Relative speed = (B1 + S) + (B2 - S) = B1 + B2
Time to meet = D / (B1 + B2)
Distance from A = (B1 + S) x D / (B1 + B2)
9. Floating Object Problems
A common problem type: a boat drops an object in the river and later turns back to retrieve it.
Scenario:
- Boat goes upstream, drops a hat at point P.
- Continues upstream for time T, then turns back.
- After how long (from turning back) does the boat reach the hat?
KEY INSIGHT:
From the hat's reference frame (floating with the current), the boat
goes away at speed B (upstream, relative to water) for time T, then
comes back at speed B (downstream, relative to water).
So the boat reaches the hat in EXACTLY T time after turning back.
The stream speed is IRRELEVANT to this calculation!
Why Stream Speed is Irrelevant
In the river's reference frame (moving with the water):
- The hat is stationary (it floats with the water).
- The boat moves at speed B both ways (relative to water).
- So return time = departure time = T.
This elegant solution avoids complex algebra.
10. Man Swimming in a River
The same formulas apply to a person swimming:
Swimming downstream: Effective speed = Swimming speed + Current speed
Swimming upstream: Effective speed = Swimming speed - Current speed
Special case: If the swimmer's speed equals the current speed:
Upstream speed = B - S = 0
The swimmer cannot make any progress upstream!
11. Effect of Wind on Aircraft (Analogy)
The boats and streams concept extends to aircraft with wind:
Tailwind (wind from behind): Effective speed = Aircraft speed + Wind speed
Headwind (wind from front): Effective speed = Aircraft speed - Wind speed
Crosswind: Requires vector addition (more complex)
12. Escalator Problems (Analogy)
Moving escalators follow the same logic:
Walking WITH the escalator:
Effective speed = Walking speed + Escalator speed
(measured in steps per second or steps per minute)
Walking AGAINST the escalator:
Effective speed = Walking speed - Escalator speed
13. Special Formulas
If upstream time is n times the downstream time
T_up = n x T_down
D/(B-S) = n x D/(B+S)
B + S = n(B - S)
B + S = nB - nS
S(1 + n) = B(n - 1)
B/S = (n + 1) / (n - 1)
Example:
If upstream time is 3 times downstream time:
B/S = (3+1)/(3-1) = 4/2 = 2
So B = 2S, i.e., boat speed is twice the stream speed.
If speed of boat is n times the speed of stream
B = nS
Downstream speed = nS + S = (n+1)S
Upstream speed = nS - S = (n-1)S
Time ratio (up:down) = (n+1) : (n-1)
14. Summary of All Key Formulas
+-----------------------------------------------------------+
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| Downstream speed: B + S |
| Upstream speed: B - S |
| |
| B = (downstream + upstream) / 2 |
| S = (downstream - upstream) / 2 |
| |
| Time downstream = D / (B + S) |
| Time upstream = D / (B - S) |
| |
| Round trip time = 2DB / (B^2 - S^2) |
| Round trip avg speed = (B^2 - S^2) / B |
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| Meeting time (opposite ends) = D / (B1 + B2) |
| [stream cancels out!] |
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| T_up/T_down = (B+S)/(B-S) |
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| If T_up = n.T_down: B/S = (n+1)/(n-1) |
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+-----------------------------------------------------------+