Boats and Streams with Relative Speed Logic | Banking Quant Mastery

Why This Chapter Matters

Boat questions are just relative-speed questions with current added in. Once still-water speed and stream speed are separated cleanly, upstream, downstream, ratio, and time-comparison problems become routine.

Core Ideas

Downstream speed $= b + s$ , upstream speed $= b - s$ , where $b$ is still-water speed and $s$ is stream speed.
Still-water speed $= \frac{downstream + upstream}{2}$ .
Stream speed $= \frac{downstream - upstream}{2}$ .
Round-trip questions usually ask for total time, so solve each leg separately.
If the time for equal distances upstream and downstream is given, compare speeds through reciprocals rather than forcing a long equation immediately.

High-Value Formulas

Concept	Formula / Rule
Downstream	$d = b + s$
Upstream	$u = b - s$
Still-water speed	$b = \frac{d + u}{2}$
Stream speed	$s = \frac{d - u}{2}$
Equal-distance total time	$T = \frac{x}{b + s} + \frac{x}{b - s}$

How To Approach Questions

Define still-water and stream speeds first.
Convert the verbal condition into upstream or downstream time.
For back-and-forth travel, compute both times separately and add.
If a ratio is given between upstream and downstream speeds, express both in the same variable first.

Worked Examples

Example 1

Prompt: If downstream speed is $14 km/h$ and upstream speed is $7 km/h$ , find the still-water speed.

Approach: Still-water speed $= \frac{14 + 7}{2} = 10.5 km/h$ .

Example 2

Prompt: A boat covers $42$ km downstream and returns the same distance upstream with speeds $14$ and $7$ km/h respectively. Find the total time.

Approach: Time $= \frac{42}{14} + \frac{42}{7} = 3 + 6 = 9$ hours.

Example 3

Prompt: A boat covers $160$ km downstream in half the time taken for the same distance upstream. If the downstream speed is found from $96$ km in $3$ hours, find the stream speed.

Approach: Downstream speed is $32 km/h$ . If upstream takes double the time for the same distance, then upstream speed is half, so $16 km/h$ . Stream speed $= \frac{32 - 16}{2} = 8 km/h$ .

Common Mistakes

Using downstream formula for the upstream leg.
Averaging speeds directly in a round-trip time problem.
Forgetting that stream speed is half the difference between downstream and upstream speeds.
Mixing still-water speed with downstream speed inside the same step.

Quick Revision

Boat questions become stable once every sentence is rewritten in terms of still-water speed, stream speed, and direction.

9. Boats and Streams with Relative Speed Logic