Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is
Question:
Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is
Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is
Options
Answer: 15
Explanation:
Let the speed of train A = vₐ and the speed of train B = v_b. Distance between stations X and Y = D. Train A takes 10 minutes from X to Y ⇒ D = 10vₐ They meet after t minutes. Distance covered by A till meeting = vₐt Distance covered by B till meeting = v_bt At meeting point: vₐt + v_bt = D = 10vₐ ⇒ t(vₐ + v_b) = 10vₐ ⇒ t = 10vₐ / (vₐ + v_b) … (1) After meeting, train B takes 9 minutes to reach X, so distance left for B = vₐt ⇒ v_b × 9 = vₐt ⇒ t = 9v_b / vₐ … (2) Equating (1) and (2): 10vₐ / (vₐ + v_b) = 9v_b / vₐ ⇒ 10vₐ² = 9v_b(vₐ + v_b) ⇒ 10vₐ² = 9vₐv_b + 9v_b² ⇒ 10vₐ² - 9vₐv_b - 9v_b² = 0 Divide by v_b²: 10(vₐ/v_b)² - 9(vₐ/v_b) - 9 = 0 Let r = vₐ/v_b ⇒ 10r² - 9r - 9 = 0 Solving, r = [9 ± √(81 + 360)] / 20 = [9 ± 21] / 20 ⇒ r = 3/2 or r = -3/5 Discard the negative value. So, vₐ/v_b = 3/2 Then, D = 10vₐ = 10 × (3/2)v_b = 15v_b Total time for train B = D / v_b = 15 minutes Final Answer: 15 minutes
Explanation:
Let the speed of train A = vₐ and the speed of train B = v_b. Distance between stations X and Y = D. Train A takes 10 minutes from X to Y ⇒ D = 10vₐ They meet after t minutes. Distance covered by A till meeting = vₐt Distance covered by B till meeting = v_bt At meeting point: vₐt + v_bt = D = 10vₐ ⇒ t(vₐ + v_b) = 10vₐ ⇒ t = 10vₐ / (vₐ + v_b) … (1) After meeting, train B takes 9 minutes to reach X, so distance left for B = vₐt ⇒ v_b × 9 = vₐt ⇒ t = 9v_b / vₐ … (2) Equating (1) and (2): 10vₐ / (vₐ + v_b) = 9v_b / vₐ ⇒ 10vₐ² = 9v_b(vₐ + v_b) ⇒ 10vₐ² = 9vₐv_b + 9v_b² ⇒ 10vₐ² - 9vₐv_b - 9v_b² = 0 Divide by v_b²: 10(vₐ/v_b)² - 9(vₐ/v_b) - 9 = 0 Let r = vₐ/v_b ⇒ 10r² - 9r - 9 = 0 Solving, r = [9 ± √(81 + 360)] / 20 = [9 ± 21] / 20 ⇒ r = 3/2 or r = -3/5 Discard the negative value. So, vₐ/v_b = 3/2 Then, D = 10vₐ = 10 × (3/2)v_b = 15v_b Total time for train B = D / v_b = 15 minutes Final Answer: 15 minutes
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