Suresh travels 720 km to his home, partly by train and partly by car. He takes…

Concept: When a fixed total distance is split between two vehicles in two different ways, and the total travel time is known for each split, treat the reciprocal of each vehicle's speed as an unknown. Each condition then gives one linear equation of the form (distance by train / train speed) + (distance by car / car speed) = total time; solving the resulting pair of linear equations by elimination gives both speeds.

Application: Let the train's speed be x km/h and the car's speed be y km/h, and let a = 1/x and b = 1/y.

1. First condition -- 300 km by train and the remaining 720 - 300 = 420 km by car, taking 9 hours 40 minutes = 29/3 hours: 300a + 420b = 29/3 ... (i)

2. Second condition -- 450 km by train and the remaining 720 - 450 = 270 km by car, taking 50 minutes more, i.e. 10 hours 30 minutes = 21/2 hours: 450a + 270b = 21/2 ... (ii)

3. Multiply (i) by 3 and (ii) by 2 to clear denominators: 900a + 1260b = 29 ... (iii) and 900a + 540b = 21 ... (iv)

4. Subtract (iv) from (iii): 720b = 8, so b = 1/90, giving car speed y = 1/b = 90 km/h.

5. Substitute b = 1/90 into (ii): 450a + 270(1/90) = 21/2, so 450a + 3 = 10.5, so 450a = 7.5, so a = 1/60, giving train speed x = 1/a = 60 km/h.

Cross-check: with train = 60 km/h and car = 90 km/h --

1. First condition: 300/60 + 420/90 = 5 + 4.667 = 9.667 hours = 9 hours 40 minutes, matching exactly.

2. Second condition: 450/60 + 270/90 = 7.5 + 3 = 10.5 hours = 10 hours 30 minutes, which is 50 minutes more than 9 hours 40 minutes, matching exactly.

Both conditions check out, confirming the derived speeds: train = 60 km/h, car = 90 km/h.