Arvind travels from town $A$ to town $B$ , and Surbhi from town B to town $A$ , both starting at the same time along the same route. After meeting each other, Arvind takes $6$ hours to reach town $B$ while Surbhi takes $24$ hours to reach town $A$ . If Arvind travelled at a speed of $54 \mathrm{~km} / \mathrm{h}$ , then the distance, in km, between town $A$ and town $B$ is

Entered answer:

✅ Correct Answer: 972

Arvind and Surbhi start from opposite towns at the same time and travel toward each other. After they meet, Arvind takes 6 more hours to reach town B, Surbhi takes 24 more hours to reach town A, and Arvind's speed is 54 km/h.

We need to find the total distance between the towns.

When two people start from opposite ends and meet, there's a beautiful relationship:

$\frac{\text{Speed of A}}{\text{Speed of B}} = \sqrt{\frac{\text{Time B takes after meeting}}{\text{Time A takes after meeting}}}$

After meeting, each person travels the distance that the other person had already covered. Since distance = speed × time, we can derive this relationship.

Using our formula:

$\frac{S_a}{S_b} = \sqrt{\frac{T_b}{T_a}}$

Where:

$S_a$ = Arvind's speed = 54 km/h

$S_b$ = Surbhi's speed = ?

$T_a$ = Time Arvind takes after meeting = 6 hours

$T_b$ = Time Surbhi takes after meeting = 24 hours

$\frac{54}{S_b} = \sqrt{\frac{24}{6}} = \sqrt{4} = 2$

Therefore: $S_b = \frac{54}{2} = 27$ km/h

After meeting, each person travels the distance the other person had already covered.

Distance Arvind travels after meeting = Distance Surbhi had covered before meeting

= $54 \times 6 = 324$ km

Distance Surbhi travels after meeting = Distance Arvind had covered before meeting

= $27 \times 24 = 648$ km

Total distance = $324 + 648 = 972$ km

Answer: 972 km

This type of problem appears frequently in competitive exams. The formula $\frac{S_a}{S_b} = \sqrt{\frac{T_b}{T_a}}$ is your shortcut to solve these problems in under 2 minutes!

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