A certain long-distance train is operated as follows.
<1> Every day the train leaves A station at 12:00 noon and arrives at B station at 9:00 of the next morning.
<2> The train which arrived at B station leaves B station in the afternoon of the day and returns to A station.
<2> The train which arrived at B station leaves B station in the afternoon of the day and returns to A station.
<3> Since the speed of return is slower than the speed of going by 5 km/h, time for returning takes one hour and 30 minutes more than time for going.
The speed of the train going back and forth is fixed respectively.
Answer the following questions.
(1) Find the distance from A station to B station.
(2) The returning train which left B station passed by the train of going to B at 12:00 night time.
Furthermore, it passed by the another train of going to B by the time it arrived at A station.
Find the distance from A station of the second passing point.
Answer by omitting below a decimal point.
Answer
(1) 1575 km
(2) 31 km
Solution
(1) Between A and B, it takes (24 - 12) + 9 = 21 hours for going and 21 + 1.5 = 22.5 hours for returning.
This time ratio is 21 : 22.5 = 14 : 15.
The speed ratio of going and return turns into an inverse ratio of this time ratio and it is 15 : 14.
15 - 14 = 1 which is a difference of this speed ratio is equivalent to 5 km/h which is a difference of actual speed.
Since the speed of going is 15 in the value of a ratio, actual speed is 5 km/h × 15= 75 km/h.
Therefore, the distance from A station to B station is 75 × 21 = 1575 km.
(2) Although there is no reference on the time when the returning train left B station, a diagram as shown in Fig. 1 can be written from the sentence.
It passed by the going train at 12:00 night (24:00 of Fig. 1).
Since the going train just ran for 12 hours, the passing point is a 75 × 12 = 900 km from A station.
The speed of the train going back and forth is fixed respectively.
Answer the following questions.
(1) Find the distance from A station to B station.
(2) The returning train which left B station passed by the train of going to B at 12:00 night time.
Furthermore, it passed by the another train of going to B by the time it arrived at A station.
Find the distance from A station of the second passing point.
Answer by omitting below a decimal point.
Answer
(1) 1575 km
(2) 31 km
Solution
(1) Between A and B, it takes (24 - 12) + 9 = 21 hours for going and 21 + 1.5 = 22.5 hours for returning.
This time ratio is 21 : 22.5 = 14 : 15.
The speed ratio of going and return turns into an inverse ratio of this time ratio and it is 15 : 14.
15 - 14 = 1 which is a difference of this speed ratio is equivalent to 5 km/h which is a difference of actual speed.
Since the speed of going is 15 in the value of a ratio, actual speed is 5 km/h × 15= 75 km/h.
Therefore, the distance from A station to B station is 75 × 21 = 1575 km.
(2) Although there is no reference on the time when the returning train left B station, a diagram as shown in Fig. 1 can be written from the sentence.
It passed by the going train at 12:00 night (24:00 of Fig. 1).
Since the going train just ran for 12 hours, the passing point is a 75 × 12 = 900 km from A station.
The returning train passed by another going train again by the time it reaches A station.
The point of the returning train at 12:00 noon is the place where it ran from the first passing point at 12:00 night for 12 hours.
The point of the returning train at 12:00 noon is the place where it ran from the first passing point at 12:00 night for 12 hours.
The distance from A station to the second passing point is 900 - (75 - 5) × 12 = 60 km.
Since the speed ratio of the train of going and return is 15 : 14, the distance from A to the passing point is 60 x 15 / (15+14) = 31.03--- km according to Fig. 2.
Omitting below a decimal point, it is 31 km from A station.
Since the speed ratio of the train of going and return is 15 : 14, the distance from A to the passing point is 60 x 15 / (15+14) = 31.03--- km according to Fig. 2.
Omitting below a decimal point, it is 31 km from A station.
