The distance from $B$ to $C$ is thrice that from $A$ to $B$ . Two trains travel from $A$ to $C$ via $B$ . The speed of train $2$ is double that of train $1$ while traveling from $A$ to $B$ and their speeds are interchanged while traveling from $B$ to $C$ . The ratio of the time taken by train $1$ to that taken by train $2$ in travelling from $A$ to $C$ is

$7:5$

$1:4$

$5:7$

$4:1$

✅ Correct Option: 3

Route	Train 1 Speed	Train 2 Speed
A → B	$s$	$2s$
B → C	$2s$	$s$

Time = Distance ÷ Speed

From A to B:

Train 1: Time = $\frac{d}{s}$

Train 2: Time = $\frac{d}{2s}$

From B to C:

Train 1: Time = $\frac{3d}{2s}$

Train 2: Time = $\frac{3d}{s}$

Train 1 total time:

$\dfrac{d}{s} + \dfrac{3d}{2s}$

$= \dfrac{2d}{2s} + \dfrac{3d}{2s}$

$= \dfrac{5d}{2s}$

Train 2 total time:

$\dfrac{d}{2s} + \dfrac{3d}{s}$

$= \dfrac{d}{2s} + \dfrac{6d}{2s}$

$= \dfrac{7d}{2s}$

Notice that both total times have the same denominator (2s) and the same distance factors. This means we can directls compare the numerators:

Train 1: $\frac{5d}{2s}$

Train 2: $\frac{7d}{2s}$

Since the denominators are identical, the ratio is simpls $5:7$ .

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