Question

Venkat can row a boat in still water at the speed of $12\text{ km/h}$. He ferries tourists $15\text{ km}$ upstream and $18\text{ km}$ downstream in $3$ hours. Find the speed of the stream.

Hint:

Always simplify your algebraic equations by dividing out any common factors (like dividing by 3 here) before expanding.
This keeps the quadratic coefficients small and easy to factorize!

Answer 1

Step 1: Understanding the Question:
We are given the speed of Venkat's boat in still water as $12\text{ km/h}$.
The total time taken to row $15\text{ km}$ upstream and $18\text{ km}$ downstream is $3\text{ hours}$.
We need to find the speed of the water stream.

Step 2: Key Formula or Approach:
Let the speed of the stream be $x\text{ km/h}$ (where $x \lt 12$).
- Upstream speed $= 12 - x\text{ km/h}$
- Downstream speed $= 12 + x\text{ km/h}$
Using the relation $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$, we form the equation:
\[ \text{Time}_{\text{upstream}} + \text{Time}_{\text{downstream}} = \text{Total Time} \]
\[ \frac{15}{12 - x} + \frac{18}{12 + x} = 3 \]

Step 3: Detailed Explanation:

• Formulate the algebraic equation:
\[ \frac{15}{12 - x} + \frac{18}{12 + x} = 3 \]
Divide the entire equation by 3 to simplify the coefficients:
\[ \frac{5}{12 - x} + \frac{6}{12 + x} = 1 \]

• Take a common denominator to combine the fractions:
\[ \frac{5(12 + x) + 6(12 - x)}{(12 - x)(12 + x)} = 1 \]
\[ \frac{60 + 5x + 72 - 6x}{144 - x^2} = 1 \]
\[ \frac{132 - x}{144 - x^2} = 1 \]

• Set up a quadratic equation by cross-multiplying:
\[ 132 - x = 144 - x^2 \]
\[ x^2 - x - 12 = 0 \]

• Solve the quadratic equation by splitting the middle term:
\[ x^2 - 4x + 3x - 12 = 0 \]
\[ x(x - 4) + 3(x - 4) = 0 \]
\[ (x - 4)(x + 3) = 0 \]
This gives:
\[ x = 4 \quad \text{or} \quad x = -3 \]
Since speed cannot be negative, we reject $x = -3$.


Step 4: Final Answer:
The speed of the stream is $4\text{ km/h}$.