Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
Question:
Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
Two places A and B are 45 kms apart and connected by a straight road. Anil goes from A to B while Sunil goes from B to A. Starting at the same time, they cross each other in exactly 1 hour 30 minutes. If Anil reaches B exactly 1 hour 15 minutes after Sunil reaches A, the speed of Anil, in km per hour, is
Options
Answer: 12
Explanation:
Let Anil’s speed = a km/h and Sunil’s speed = s km/h. Distance AB = 45 km. They meet after 1 hour 30 minutes = 1.5 hours, so a·1.5 + s·1.5 = 45 ⇒ a + s = 45 / 1.5 = 30. Time for Anil to travel whole distance = 45 / a. Time for Sunil to travel whole distance = 45 / s. Given Anil arrives 1 hour 15 minutes = 1.25 hours after Sunil: 45 / a = 45 / s + 1.25. Divide the last equation by 45: 1 / a = 1 / s + 1.25 / 45 = 1 / s + 1/36. So (s − a) / (a s) = 1/36. Let s − a = D. Since a + s = 30, product a s = ( (a + s)^2 − (s − a)^2 ) / 4 = (900 − D^2) / 4. Then (s − a) / (a s) = D / ((900 − D^2)/4) = 4D / (900 − D^2) = 1/36. Solve: 4D·36 = 900 − D^2 ⇒ 144D = 900 − D^2 ⇒ D^2 + 144D − 900 = 0. Discriminant = 144^2 + 4·900 = 24336 = 156^2. So D = (−144 ± 156)/2. Positive root: D = (12)/2 = 6. s − a = 6 and a + s = 30 ⇒ 2a + 6 = 30 ⇒ a = 12. Answer: 12 km/h
Explanation:
Let Anil’s speed = a km/h and Sunil’s speed = s km/h. Distance AB = 45 km. They meet after 1 hour 30 minutes = 1.5 hours, so a·1.5 + s·1.5 = 45 ⇒ a + s = 45 / 1.5 = 30. Time for Anil to travel whole distance = 45 / a. Time for Sunil to travel whole distance = 45 / s. Given Anil arrives 1 hour 15 minutes = 1.25 hours after Sunil: 45 / a = 45 / s + 1.25. Divide the last equation by 45: 1 / a = 1 / s + 1.25 / 45 = 1 / s + 1/36. So (s − a) / (a s) = 1/36. Let s − a = D. Since a + s = 30, product a s = ( (a + s)^2 − (s − a)^2 ) / 4 = (900 − D^2) / 4. Then (s − a) / (a s) = D / ((900 − D^2)/4) = 4D / (900 − D^2) = 1/36. Solve: 4D·36 = 900 − D^2 ⇒ 144D = 900 − D^2 ⇒ D^2 + 144D − 900 = 0. Discriminant = 144^2 + 4·900 = 24336 = 156^2. So D = (−144 ± 156)/2. Positive root: D = (12)/2 = 6. s − a = 6 and a + s = 30 ⇒ 2a + 6 = 30 ⇒ a = 12. Answer: 12 km/h
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