A boat goes $30$ km upstream and $44$ km downstream in $10$ hours. In $13$ hours, it can go $40$ km upstream and $55$ km downstream. Determine the speed of the stream and that of the boat in still water.
Step-by-Step Solution
Key Concept: Let speed of boat in still water be $x$ km/h and speed of stream be $y$ km/h. Upstream speed $= x - y$, downstream speed $= x + y$. Let $u = \dfrac{1}{x-y}$ and $v = \dfrac{1}{x+y}$.
Upstream speed $= x - y$, downstream speed $= x + y$.
Time $=$ Distance / Speed.
Equation 1: $\dfrac{30}{x-y} + \dfrac{44}{x+y} = 10 \Rightarrow 30u + 44v = 10$.
Equation 2: $\dfrac{40}{x-y} + \dfrac{55}{x+y} = 13 \Rightarrow 40u + 55v = 13$. [1.5 Marks]
Multiply Eq 1 by 4: $120u + 176v = 40$.
Multiply Eq 2 by 3: $120u + 165v = 39$.
Subtract: $11v = 1 \Rightarrow v = \dfrac{1}{11}$.
Then $30u + 44\left(\dfrac{1}{11}\right) = 10 \Rightarrow 30u + 4 = 10 \Rightarrow 30u = 6 \Rightarrow u = \dfrac{1}{5}$. [1.5 Marks]
We have $x - y = \dfrac{1}{u} = 5$ -- (eq 3) and $x + y = \dfrac{1}{v} = 11$ -- (eq 4). [1.0 Mark]
Add eq 3 and eq 4: $2x = 16 \Rightarrow x = 8$ km/h (speed of boat in still water).
Subtract eq 3 from eq 4: $2y = 6 \Rightarrow y = 3$ km/h (speed of stream). [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Setting up upstream/downstream speed and time equations in $u, v$: 1.5 Marks
Solving linear system for $u = 1/5, v = 1/11$: 1.5 Marks
Forming system for $x-y=5$ and $x+y=11$: 1.0 Mark
Final values: Speed of boat $= 8$ km/h, Speed of stream $= 3$ km/h: 1.0 Mark
Correct Answer: