Exercise 3.3 – Question 4: Boat and Stream Problem
📚 Understanding Boat and Stream Problems
Key Concepts:
- Upstream: Boat moves against the stream (current opposes motion)
- Downstream: Boat moves with the stream (current aids motion)
- Speed in still water: Speed of boat when there’s no current
- Speed of stream: Speed of water current
📐 Important Formulas
Let:
- \( u \) = Speed of boat in still water (km/h)
- \( v \) = Speed of stream (km/h)
Then:
- Upstream speed = \( u – v \) km/h (boat speed – stream speed)
- Downstream speed = \( u + v \) km/h (boat speed + stream speed)
- Time = \( \frac{\text{Distance}}{\text{Speed}} \)
Visual Understanding
🚤 Boat Motion: Upstream vs Downstream
Visual representation of boat moving upstream and downstream
Step 1: Define Variables
Speed of boat in still water = \( u \) km/h
Speed of stream = \( v \) km/h
Then:
Upstream speed = \( (u - v) \) km/h
Downstream speed = \( (u + v) \) km/h
Step 2: Form Equations Using Time Formula
Time for upstream = \( \frac{30}{u - v} \) hours
Time for downstream = \( \frac{44}{u + v} \) hours
Total time = 10 hours
\[ \frac{30}{u - v} + \frac{44}{u + v} = 10 \quad \text{...(1)} \]Time for upstream = \( \frac{40}{u - v} \) hours
Time for downstream = \( \frac{55}{u + v} \) hours
Total time = 13 hours
\[ \frac{40}{u - v} + \frac{55}{u + v} = 13 \quad \text{...(2)} \]Step 3: Simplify by Substitution
Let \( \frac{1}{u - v} = x \) (reciprocal of upstream speed)
Let \( \frac{1}{u + v} = y \) (reciprocal of downstream speed)
Then equations become:
Equation (1): \( 30x + 44y = 10 \) ...(3)
Equation (2): \( 40x + 55y = 13 \) ...(4)
Step 4: Solve Using Elimination Method
Equation (3) × 4: \( 120x + 176y = 40 \) ...(5)
Equation (4) × 3: \( 120x + 165y = 39 \) ...(6)
Step 5: Find u and v
\( x = \frac{1}{u - v} = \frac{1}{5} \)
Therefore: \( u - v = 5 \) ...(7)
\( y = \frac{1}{u + v} = \frac{1}{11} \)
Therefore: \( u + v = 11 \) ...(8)
Step 6: Verification
Upstream speed = \( 8 - 3 = 5 \) km/h
Downstream speed = \( 8 + 3 = 11 \) km/h
Check condition 1:
Time for 30 km upstream = \( \frac{30}{5} = 6 \) hours
Time for 44 km downstream = \( \frac{44}{11} = 4 \) hours
Total time = \( 6 + 4 = 10 \) hours ✓
Check condition 2:
Time for 40 km upstream = \( \frac{40}{5} = 8 \) hours
Time for 55 km downstream = \( \frac{55}{11} = 5 \) hours
Total time = \( 8 + 5 = 13 \) hours ✓
Visual Summary
📊 Speed Comparison Chart
Bar chart comparing different speeds
Speed of boat in still water = 8 km/h
Speed of stream = 3 km/h
📊 Summary Table
| Type | Formula | Value |
|---|---|---|
| Boat in still water | \( u \) | 8 km/h |
| Stream speed | \( v \) | 3 km/h |
| Upstream speed | \( u - v \) | 5 km/h |
| Downstream speed | \( u + v \) | 11 km/h |
⚠️ Common Mistakes to Avoid
✅ Correct: Upstream = \( u - v \) (slower), Downstream = \( u + v \) (faster)
✅ Correct: Always use this formula to form equations.
✅ Correct: Let \( \frac{1}{u-v} = x \) and \( \frac{1}{u+v} = y \) to simplify.
✅ Correct: Be careful with signs, especially when subtracting.
💡 Key Points to Remember
Boat and Stream Problems
- Basic Formulas:
- Upstream speed = Boat speed - Stream speed = \( u - v \)
- Downstream speed = Boat speed + Stream speed = \( u + v \)
- Time = Distance ÷ Speed
- Finding u and v:
- If \( u - v = a \) and \( u + v = b \)
- Then: \( u = \frac{a + b}{2} \) and \( v = \frac{b - a}{2} \)
- Problem-solving strategy:
- Define variables clearly
- Use time formula to form equations
- Simplify using substitution if needed
- Solve using elimination or substitution
- Always verify the answer
- Physical understanding:
- Upstream is slower (against current)
- Downstream is faster (with current)
- Stream helps in one direction, opposes in the other
📚 Related Questions
Farhan Mansuri
M.Sc. Mathematics, B.Ed.
15+ Years Teaching Experience
Passionate about making mathematics easy and enjoyable for students.