एक कार v km/hr की औसत चाल से स्थान X से स्थान Y तक यात्रा करती है, Y से X तक 2v km/hr की औसत चाल से, फिर X से Y तक 3v km/hr की औसत चाल से और फिर Y से X तक 4v km/hr की औसत चाल से यात्रा करती है। तो संपूर्ण यात्रा के लिए कार की औसत चाल
सही उत्तर (B) (b) lies between v and 2v km/hr है।
व्याख्या:
Let the distance between place X and place Y be d km. The car makes 4 legs of the journey, each of distance d: Leg 1 (X to Y) at speed v, time t1 = d / v. Leg 2 (Y to X) at speed 2v, time t2 = d / (2v). Leg 3 (X to Y) at speed 3v, time t3 = d / (3v). Leg 4 (Y to X) at speed 4v, time t4 = d / (4v). Total distance for the entire journey = 4d. Total time taken T = t1 + t2 + t3 + t4 = (d/v) + (d/2v) + (d/3v) + (d/4v) = (d/v) × [1 + 1/2 + 1/3 + 1/4]. The term inside the bracket = (12 + 6 + 4 + 3) / 12 = 25 / 12. Thus, T = (d/v) × (25/12) = 25d / (12v). Average speed = Total Distance / Total Time = 4d / [25d / (12v)] = 4 × 12v / 25 = 48v / 25 = 1.92v. Since 1.92v lies between 2v and 3v? Wait, 1.92v is less than 2v, so it lies between v and 2v. Let us check: 1.92 is between 1 and 2, so the average speed lies between v and 2v. Option (b) is correct.
In English (Question & Model Answer)
A car travels from a place X to place Y at an average speed of v km/hr, from Y to X at an average speed of 2v km/hr, again from X to Y at an average speed of 3v km/hr and again from Y to X at an average speed of 4v km/hr. Then the average speed of the car for the entire journey
The correct answer is (B) (b) lies between v and 2v km/hr.
Explanation:
Let the distance between place X and place Y be d km. The car makes 4 legs of the journey, each of distance d: Leg 1 (X to Y) at speed v, time t1 = d / v. Leg 2 (Y to X) at speed 2v, time t2 = d / (2v). Leg 3 (X to Y) at speed 3v, time t3 = d / (3v). Leg 4 (Y to X) at speed 4v, time t4 = d / (4v). Total distance for the entire journey = 4d. Total time taken T = t1 + t2 + t3 + t4 = (d/v) + (d/2v) + (d/3v) + (d/4v) = (d/v) × [1 + 1/2 + 1/3 + 1/4]. The term inside the bracket = (12 + 6 + 4 + 3) / 12 = 25 / 12. Thus, T = (d/v) × (25/12) = 25d / (12v). Average speed = Total Distance / Total Time = 4d / [25d / (12v)] = 4 × 12v / 25 = 48v / 25 = 1.92v. Since 1.92v lies between 2v and 3v? Wait, 1.92v is less than 2v, so it lies between v and 2v. Let us check: 1.92 is between 1 and 2, so the average speed lies between v and 2v. Option (b) is correct.