9. Boats and Streams with Relative Speed Logic

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 $=\frac{\text{downstream}+\text{upstream}}{2}$.
• Stream speed $=\frac{\text{downstream}-\text{upstream}}{2}$.
• 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

How To Approach Questions

1. Define still-water and stream speeds first.
2. Convert the verbal condition into upstream or downstream time.
3. For back-and-forth travel, compute both times separately and add.
4. 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 $14\text{ km/h}$ and upstream speed is $7\text{ km/h}$, find the still-water speed.

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

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

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.