Two trains A and B started from stations P and Q respectively towards each other. Train A started at 7 p.m. at a speed of 60 km/hr and train B started at 4 a.m. (next day) at a speed of 90 km/hr. The distance between the two stations P and Q is 800 km.
How far from station Q will the two trains meet?
This problem requires us to determine the exact location where two trains, Train A and Train B, will cross paths. They are traveling towards each other from different stations, P and Q, and commence their journeys at different times. Our goal is to find the distance of this meeting point measured from station Q.
First, we must account for the time Train A travels before Train B starts. Train A begins at 7 p.m., and Train B begins at 4 a.m. the next day. The duration between these two start times is 9 hours (from 7 p.m. to midnight is 5 hours, and from midnight to 4 a.m. is 4 hours).
Time Train A travels alone = 9 hours
Train A covers a certain distance during these 9 hours. We use the formula: Distance = Speed \(\times\) Time.
Distance covered by Train A = Speed of Train A \(\times\) Time Train A travels alone
Distance_A_initial = \(60 \text{ km/hr} \times 9 \text{ hr}\)
Distance_A_initial = \(540 \text{ km}\)
When Train B starts at 4 a.m., Train A has already traveled 540 km from station P. The distance remaining between the two trains is the total distance minus the distance Train A has covered.
Remaining Distance = Total Distance PQ - Distance_A_initial
Remaining Distance = \(800 \text{ km} - 540 \text{ km}\)
Remaining Distance = \(260 \text{ km}\)
At 4 a.m., the trains are 260 km apart and are moving towards each other.
Since both trains are moving towards each other, their speeds combine to close the distance between them. This combined speed is called the relative speed.
Relative Speed = Speed of Train A + Speed of Train B
Relative Speed = \(60 \text{ km/hr} + 90 \text{ km/hr}\)
Relative Speed = \(150 \text{ km/hr}\)
Now we calculate how long it will take for the trains to cover the remaining 260 km distance at their relative speed. Using the formula: Time = Distance / Speed.
Time to meet = Remaining Distance / Relative Speed
Time to meet = \(260 \text{ km} / 150 \text{ km/hr}\)
Time to meet = \(\frac{260}{150} \text{ hr} = \frac{26}{15} \text{ hr}\)
The question asks for the distance from station Q. This is the distance Train B travels from station Q until they meet. Using the formula: Distance = Speed \(\times\) Time.
Distance from Q = Speed of Train B \(\times\) Time to meet
Distance from Q = \(90 \text{ km/hr} \times \frac{26}{15} \text{ hr}\)
Distance from Q = \(\frac{90 \times 26}{15} \text{ km}\)
Distance from Q = \(6 \times 26 \text{ km}\)
Distance from Q = 156 km
Therefore, the two trains will meet 156 km away from station Q.
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