Shortcut Trick Ratio of distances covered by Kiran and Rahul = (900 − 270) : (900 − 340) ⇒ Kiran : Rahul = 630 : 560 = 9 : 8 To beat in a 900 m race, Kiran must cover 9 units = 900 m. ⇒ 1 unit = 100 m. Beating distance = Difference in ratio parts = 9 − 8 = 1 unit = 100 m. ∴ The correct answer is 100 m. Alternate Method Given: Total Race Distance = 900 m. When Sathish completes 900 m: ⇒ Distance covered by Kiran = 900 − 270 = 630 m. ⇒ Distance covered by Rahul = 900 − 340 = 560 m. The ratio of their speeds is equal to the ratio of distances covered in the same time: ⇒ Speed of Kiran / Speed of Rahul = 630 / 560 = 9 / 8 In a 900 m race, when Kiran covers 900 m, let Rahul cover distance D: ⇒ 900 / D = 9 / 8 ⇒ D = (900 × 8) ÷ 9 ⇒ D = 100 × 8 = 800 m Distance by which Kiran beats Rahul = 900 − 800 = 100 m. ∴ The correct answer is 100 m. Related Concepts: Speed and Distance Proportionality When time is constant, distance is directly proportional to speed (D_(1)/D_(2) = S_(1)/S_(2)). This is the core logic used in race problems. Concept of Beating In a race of length L, if A beats B by x metres, it means when A covers L metres, B has covered only (L − x) metres. Dead Heat A dead heat occurs when two or more contestants reach the finish line at the exact same time, meaning they cover the race distance in the same time period.