Problems on Trains - Formulas

Problem on train - Formula Table

Explanation with example for each formula

Speed of the Train:

$ \text{Speed of the Train} = \frac{\text{Total distance covered by the train}}{\text{Time taken}} $

Time taken to cross each other in opposite directions:

$ \text{Time taken} = \frac{a + b}{x + y} $

Example: Two trains with lengths 100 meters and 120 meters are moving towards each other with speeds of 60 km/h and 80 km/h, respectively. How much time will they take to cross each other?

Substituting the given values into the formula: $ \text{Time taken} = \frac{100 + 120}{60 + 80} = \frac{220}{140} = 1.57 $ hours (rounded to two decimal places).

Time taken to cross each other in the same direction:

$ \text{Time taken} = \frac{a + b}{x - y} $

Example: Two trains with lengths 150 meters and 200 meters are moving in the same direction with speeds of 40 km/h and 20 km/h, respectively. How much time will it take for the faster train to cross the slower train?

Substituting the given values into the formula: $ \text{Time taken} = \frac{150 + 200}{40 - 20} = \frac{350}{20} = 17.5 $ hours.

Ratio between the speed of two trains:

$ \text{Speed ratio} = \sqrt{\frac{t_2}{t_1}} $

Example: Two trains start simultaneously from stations X and Y towards each other. After crossing each other, they took 4 hours and 9 hours to reach stations Y and X, respectively. Find the ratio between their speeds.

Substituting the given values into the formula: $ \text{Speed ratio} = \sqrt{\frac{9}{4}} = \sqrt{2.25} = 1.5 $.

Distance from starting point where two trains meet:

$ \text{Distance} = (t_2 - t_1) \times \frac{\text{Product of speeds}}{\text{Difference in speeds}} $

Example: Two trains leave stations X and Y at times $ t_1 $ and $ t_2 $, respectively. The speeds of the trains are 60 km/h and 80 km/h. If they meet after 3 hours, find the distance from station X where they meet.

Substituting the given values into the formula: $ \text{Distance} = (t_2 - t_1) \times \frac{\text{Product of speeds}}{\text{Difference in speeds}} = 3 \times \frac{60 \times 80}{80 - 60} = 3 \times \frac{4800}{20} = 720 $ km.

Rest Time per hour:

$ \text{Rest Time per hour} = \frac{\text{Difference in average speed}}{\text{Speed without stoppage}} $

Example: A train covers a certain distance without any stoppage at an average speed of 80 km/h. However, with a 10-minute stoppage per hour, it covers the same distance at an average speed of 60 km/h. Find the rest time per hour.

Substituting the given values into the formula: $ \text{Rest Time per hour} = \frac{\text{Difference in average speed}}{\text{Speed without stoppage}} = \frac{80 - 60}{80} = \frac{20}{80} = 0.25 { hours}$

which is equivalent to 15 minutes.

Time taken to cross each other in opposite directions with equal lengths:

$ \text{Time taken} = \frac{2 \times t_1 \times t_2}{t_1 + t_2} $

Example: Two trains of equal lengths take 10 seconds and 15 seconds, respectively, to cross a pole. Find the time taken by them to cross each other if they are moving in opposite directions.

Substituting the given values into the formula: $ \text{Time taken} = \frac{2 \times 10 \times 15}{10 + 15} = \frac{300}{25} = 12 $ seconds.

Time taken to cross each other in the same direction with equal lengths:

$ \text{Time taken} = \frac{2 \times t_1 \times t_2}{t_2 - t_1} $

Example: Two trains of equal lengths take 12 seconds and 18 seconds, respectively, to cross a pole. Find the time taken by them to cross each other if they are moving in the same direction.

Substituting the given values into the formula: $ \text{Time taken} = \frac{2 \times 12 \times 18}{18 - 12} = \frac{432}{6} = 72 $ seconds.