Trains Meeting: Calculating Distance Between Stations

Let \( v_A \) be the speed of the train starting from station A, and \( v_B \) be the speed of the train starting from station B.

• Given: \( v_A = 60 \) km/h and \( v_B = 70 \) km/h.
• The trains travel towards each other and meet after a certain time, say \( t \) hours.
• Let \( d_A \) be the distance travelled by the train from A and \( d_B \) be the distance travelled by the train from B.
• Distance is calculated as Speed × Time. So, \( d_A = v_A \times t = 60t \) and \( d_B = v_B \times t = 70t \).
• The ratio of distances travelled is equal to the ratio of their speeds:

  $ \frac{d_A}{d_B} = \frac{v_A}{v_B} = \frac{60}{70} = \frac{6}{7} $

• We can express the distances as \( d_A = 6x \) and \( d_B = 7x \) for some constant \( x \).
• We are given that the second train (from B) travelled 100 km more than the first train (from A):

  $ d_B - d_A = 100 $

• Substituting the expressions in terms of \( x \):

  $ 7x - 6x = 100 $

  $ x = 100 $

• Now, calculate the distances travelled by each train:

  $ d_A = 6x = 6 \times 100 = 600 \text{ km} $

  $ d_B = 7x = 7 \times 100 = 700 \text{ km} $

• The total distance between stations A and B is the sum of the distances travelled by both trains until they meet:

  $ \text{Distance between A and B} = d_A + d_B $

  $ \text{Distance between A and B} = 600 \text{ km} + 700 \text{ km} = 1300 \text{ km} $

Therefore, the distance between A and B is 1300 km.