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 km/h, then the distance, in km, between town A and town B is:

Question

Accepted Answer

Let the distance between town A and the meeting point M be $$ D_1 $$ and the distance between town B and the meeting point M be $$ D_2 $$.

#### Step 1: Relation between distances
Arvind covers $$ D_1 $$ in $$ t $$ hours and $$ D_2 $$ in 6 hours. Surbhi covers $$ D_1 $$ in 24 hours and $$ D_2 $$ in $$ t $$ hours. Using their speeds:  
- Speed of Arvind:  
$$ \frac{D_1}{t} = \frac{D_2}{6}. $$  
Thus:  
$$ \frac{D_1}{D_2} = \frac{t}{6}. $$

- Speed of Surbhi:  
$$ \frac{D_1}{24} = \frac{D_2}{t}. $$  
Thus:  
$$ \frac{D_1}{D_2} = \frac{24}{t}. $$

Equating both ratios:  
$$ \frac{t}{6} = \frac{24}{t}. $$  
Solve for $$ t $$:  
$$ t^2 = 6 	imes 24 = 144. $$  
$$ t = 12 	ext{ hours}. $$

#### Step 2: Total time taken by Arvind
The total time taken by Arvind to travel from town A to town B is:  
$$ t + 6 = 12 + 6 = 18 	ext{ hours}. $$

#### Step 3: Distance calculation
The speed of Arvind is given as 54 km/h. Therefore, the distance between town A and town B is:  
$$ 	ext{Distance} = 	ext{Speed} 	imes 	ext{Time} = 54 	imes 18 = 972 	ext{ km}. $$

Thus, the distance between towns A and B is **972 km**.