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Pair of Linear Equations in Two Variables
RD Sharma
CBSE
Grade 10

Question:

A train covered a certain distance at a uniform speed. If the train had been $10\text{ km/h}$ faster, it would have taken $2$ hours less than the scheduled time. And, if the train were slower by $10\text{ km/h}$; it would have taken $3$ hours more than the scheduled time. Find the distance covered by the train.

Step-by-Step Solution

Key Concept: Speed $= v$, Time $= t$, Distance $= vt$. (1) $(v+10)(t-2) = vt \Rightarrow -2v + 10t = 20 \Rightarrow -v + 5t = 10$. (2) $(v-10)(t+3) = vt \Rightarrow 3v - 10t = 30$. Multiply (1) by 2: $-2v + 10t = 20$. Add to (2): $v = 50\text{ km/h} \Rightarrow t = 12\text{ hours}$. Distance $= 50 \times 12 = 600\text{ km}$.
Equations: (1) $-v + 5t = 10$, (2) $3v - 10t = 30$. [1.0 Mark]
Multiply (1) by 2: $-2v + 10t = 20$. Add to (2): $v = 50\text{ km/h}$. [1.0 Mark]
$t = (50 + 10)/5 = 12\text{ hours}$. Distance $= 50 \times 12 = 600\text{ km}$. [1.0 Mark]

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🎯 Official CBSE Marking Scheme:
Forming speed-time equations: 1.0 Mark
Solving $v = 50\text{ km/h}$ and $t = 12\text{ hours}$: 1.0 Mark
Evaluating distance $= 600\text{ km}$: 1.0 Mark

Correct Answer:
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