Form the pair of linear equations for the following problem and solve it algebraically:
Roohi travels $300\text{ km}$ to her home partly by train and partly by bus. She takes $4$ hours if she travels $60\text{ km}$ by train and the remaining by bus. If she travels $100\text{ km}$ by train and the remaining by bus, she takes $10$ minutes longer. Find the speed of the train and the bus separately.
Step-by-Step Solution
Key Concept: Speed of train $= u$, speed of bus $= v$. (1) $60/u + 240/v = 4$. (2) $100/u + 200/v = 4 + 1/6 = 25/6$. Let $x = 1/u, y = 1/v$. $60x + 240y = 4 \Rightarrow 15x + 60y = 1$. $100x + 200y = 25/6 \Rightarrow 24x + 48y = 1$. Multiply 1st by 8, 2nd by 5: $120x + 480y = 8$ and $120x + 240y = 5$. Subtract: $240y = 3 \Rightarrow y = 1/80 \Rightarrow v = 80\text{ km/h}$. $15x + 60(1/80) = 1 \Rightarrow 15x = 1/4 \Rightarrow x = 1/60 \Rightarrow u = 60\text{ km/h}$.
Equations with $x = 1/u, y = 1/v$: (1) $15x + 60y = 1$, (2) $24x + 48y = 1$. [1.5 Marks]
Multiply (1) by 8, (2) by 5: $120x + 480y = 8$ and $120x + 240y = 5$. Subtract: $240y = 3 \Rightarrow y = 1/80 \Rightarrow v = 80\text{ km/h}$. [2.0 Marks]
$15x + 60(1/80) = 1 \Rightarrow 15x = 1/4 \Rightarrow x = 1/60 \Rightarrow u = 60\text{ km/h}$.
Speed of train $= 60\text{ km/h}$, Speed of bus $= 80\text{ km/h}$. [1.5 Marks]
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🎯 Official CBSE Marking Scheme:
Forming linear equations in $1/u, 1/v$: 1.5 Marks
Solving speed of bus $v = 80\text{ km/h}$: 2.0 Marks
Solving speed of train $u = 60\text{ km/h}$: 1.5 Marks
Correct Answer: