A goods train and a passenger train are running on parallel tracks in the same direction. The driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 90 sec, whereas a passenger on the passenger train marks that he crosses the goods train in 40 sec. If the speeds of the trains are in the ratio 1:2, find the ratio of their lengths?
Question
A goods train and a passenger train are running on parallel tracks in the same direction. The driver of the goods train observes that the passenger train coming from behind overtakes and crosses his train completely in 90 sec, whereas a passenger on the passenger train marks that he crosses the goods train in 40 sec. If the speeds of the trains are in the ratio 1:2, find the ratio of their lengths?
Solution
Let's denote:
- L1 = length of the goods train
- L2 = length of the passenger train
- S1 = speed of the goods train
- S2 = speed of the passenger train
From the problem, we know that the speeds of the trains are in the ratio 1:2. So, we can write S2 = 2*S1.
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When the passenger train overtakes the goods train, they travel the distance equal to the sum of their lengths at the relative speed (S2 - S1). The time it takes is 90 seconds. So, we can write the first equation: (L1 + L2) / (S2 - S1) = 90.
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When a passenger on the passenger train observes that he crosses the goods train, they travel the distance equal to the length of the goods train at the relative speed (S2). The time it takes is 40 seconds. So, we can write the second equation: L1 / S2 = 40.
Substituting S2 = 2*S1 into the equations, we get:
- (L1 + L2) / S1 = 180 (from the first equation)
- L1 / (2*S1) = 40 (from the second equation)
From the second equation, we can express L1 = 80S1. Substituting this into the first equation, we get L2 = 100S1.
So, the ratio of their lengths L1:L2 is 80S1 : 100S1 = 4:5.
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