Problems On Trains
Name: _____________________
Date: _____________________
Instructions: Answer all questions. Write your answers clearly in the space provided.
The Ghaziabad - Hapur - Meerut EMU and the Meerut - Hapur - Ghaziabad EMU start at the same time from Ghaziabad and Meerut and proceed towards each other at 16 km/hr and 21 km/hr respectively. When they meet, it is found that one train has traveled 60 km more than the other . The distance between two stations is?
Two trains, A ans B start from stations X and Y towards each other, they take 4 hours 48 minutes and 3 hours 20 minutes to reach Y and X respectively after they meet. If train A is moving at 45 km/hr, then the speed of the train B is?
A train passes by a lamp post at platform in 7 sec. and passes by the platform completely in 28 sec. If the length of the platform is 390m, then length of the train (in meters) is?
A train moving at a rate of 36 km/hr crosses a standing man in 10 seconds. It will cross a platform 55 meters long in?
Two trains start at the same time for two station A and B toward B and A respectively. If the distance between A and B is 220 km and their speeds are 50 km/hr and 60 km/hr respectively then after how much time will they meet each other?
A train 100 meter long meets a man going in opposite direction at 5 km/h and passes him in 7 1 / 5 seconds. What is the speed of the train (in km/hr)?
A train takes 9 sec to cross a pole. If the speed of the train is 48 kmph, then length of the train is?
Two trains start at the same time from A and B and proceed toward each other at the sped of 75 km/hr and 50 km/hr respectively. When both meet at a point in between, one train was found to have traveled 175 km more then the other. Find the distance between A and B?
Two trains 180 meters and 120 meters in length are running towards each other on parallel tracks, one at the rate 65 km/hr and another at 55 km/hr. In how many seconds will they be cross each other from the moment they meet?
Two train 100 meters and 95 meters long respectively pass each other in 27 seconds, when they run in the same direction and in 9 seconds when they run in opposite directions. Speed of the two trains are?
A train running at the speed of 84 km/hr passes a man walking in opposite direction at the speed of 6 km/hr in 4 seconds. What is the length of train (in meter)?
A train passes two bridges of length 500 m and 250 m in 100 seconds and 60 seconds respectively. The length of the train is?
Train A passes a lamp post in 3 seconds and 900 meter long platform in 30 seconds. How much time will the same train take to cross a platform which is 800 meters long? (in seconds)
A train cover a distance of 3584 km in 2 days 8 hours. If it covers 1440 km on the first day and 1608 km on the second day, by how much does the average speed of the train for the remaining part of the journey differ from that for the entire journey?
A train starts from A at 7 a.m. towards B with speed 50 km/h. Another train starts from B at 8 a.m. with speed of 60 km/h towards A. Both of them meet at 10 a.m. at C. The ratio of the distance AC to BC is?
Train A passes a lamp post in 9 seconds and 700 meter long platform in 30 seconds. How much time will the same train take to cross a platform which is 800 meters long? (in seconds)
Train A traveling at 63 kmph can cross a platform 199.5 m long in 21 seconds. How much would train A take to completely cross (from the moment they meet ) train B, 157 m long and traveling at 54 kmph in opposite direction which train A is traveling? (in seconds)
A train which is moving at an average speed of 40 km/h reaches its destination on time. When its average speed reduces to 35 km/h, then it reaches its destination 15 minutes late. The distance traveled by the train is?
A train moves with a speed of 30 kmph for 12 minutes and for next 8 minutes at a speed of 45 kmph. Find the average speed of the train?
A train running at the speed of 60 km/hr crosses a pole in 9 seconds. What is the length of the train?
A train 125 m long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:
The length of the bridge, which a train 130 metres long and travelling at 45 km/hr can cross in 30 seconds, is:
Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:
A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform?
A train 240 m long passes a pole in 24 seconds. How long will it take to pass a platform 650 m long?
Two trains of equal length are running on parallel lines in the same direction at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is:
A train 360 m long is running at a speed of 45 km/hr. In what time will it pass a bridge 140 m long?
Two trains are moving in opposite directions @ 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in seconds is:
A jogger running at 9 kmph alongside a railway track in 240 metres ahead of the engine of a 120 metres long train running at 45 kmph in the same direction. In how much time will the train pass the jogger?
How many seconds will a 500 metre long train take to cross a man walking with a speed of 3 km/hr in the direction of the moving train if the speed of the train is 63 km/hr?
Two goods train each 500 m long, are running in opposite directions on parallel tracks. Their speeds are 45 km/hr and 30 km/hr respectively. Find the time taken by the slower train to pass the driver of the faster one.
Two trains are running in opposite directions with the same speed. If the length of each train is 120 metres and they cross each other in 12 seconds, then the speed of each train (in km/hr) is:
Two trains of equal lengths take 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 metres, in what time (in seconds) will they cross each other travelling in opposite direction?
A train 108 m long moving at a speed of 50 km/hr crosses a train 112 m long coming from opposite direction in 6 seconds. The speed of the second train is:
Two trains are running at 40 km/hr and 20 km/hr respectively in the same direction. Fast train completely passes a man sitting in the slower train in 5 seconds. What is the length of the fast train?
A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is:
A train overtakes two persons walking along a railway track. The first one walks at 4.5 km/hr. The other one walks at 5.4 km/hr. The train needs 8.4 and 8.5 seconds respectively to overtake them. What is the speed of the train if both the persons are walking in the same direction as the train?
A train travelling at 48 kmph completely crosses another train having half its length and travelling in opposite direction at 42 kmph, in 12 seconds. It also passes a railway platform in 45 seconds. The length of the platform is
Two stations A and B are 110 km apart on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 kmph. Another train starts from B at 8 a.m. and travels towards A at a speed of 25 kmph. At what time will they meet?
A train 75 m long overtook a person who was walking at the rate of 6 km/hr in the same direction and passed him in $$7frac{1}{2}$$ seconds. Subsequently, it overtook a second person and passed him in $$6frac{3}{4}$$ seconds. At what rate was the second person travelling?
Two trains are running in opposite directions with the same speed. If the length of each train is 120 meters and they cross each other in 12 seconds, then the speed of each train (in km/hr) is?
A 150 m long train crosses a milestone in 15 seconds and a train of same length coming from the opposite direction in 12 seconds. The speed of the other train is?
A man standing on a platform finds that a train takes 3 seconds to pass him and another train of the same length moving in the opposite direction takes 4 seconds. The time taken by the trains to pass each other will be :
Two trains, 130 and 110 meters long, are going in the same direction. The faster train takes one minute to pass the other completely. If they are moving in opposite directions, they pass each other completely in 3 seconds. Find the speed of the faster train.
Two identical trains A and B running in opposite directions at same speed take 2 minutes to cross each other completely. The number of bogies of A are increased from 12 to 16. How much more time would they now require to cross each other?
A 270 metres long train running at the speed of 120 kmph crosses another train running in opposite direction at the speed of 80 kmph in 9 seconds. What is the length of the other train?
A goods train runs at the speed of 72 kmph and crosses a 250 m long platform in 26 seconds. What is the length of the goods train?
Two trains, each 100 m long, moving in opposite directions, cross each other in 8 seconds. If one is moving twice as fast the other, then the speed of the faster train is:
Two trains 140 m and 160 m long run at the speed of 60 km/hr and 40 km/hr respectively in opposite directions on parallel tracks. The time (in seconds) which they take to cross each other, is:
A train 110 metres long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
A train travelling at a speed of 75 mph enters a tunnel 3 1 / 2 miles long. The train is 1 / 4 mile long. How long does it take for the train to pass through the tunnel from the moment the front enters to the moment the rear emerges?
A train 800 metres long is running at a speed of 78 km/hr. If it crosses a tunnel in 1 minute, then the length of the tunnel (in meters) is:
A 300 metre long train crosses a platform in 39 seconds while it crosses a signal pole in 18 seconds. What is the length of the platform?
A train speeds past a pole in 15 seconds and a platform 100 m long in 25 seconds. Its length is:
A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds respectively. What is the speed of the train?
A man sitting in a train is counting the pillars of electricity. The distance between two pillars is 60 meters, and the speed of the train is 42 km/hr. In 5 hours, how many pillars will he count?
A 120 meter long train is running at a speed of 90 km/hr. It will cross a railway platform 230 m long in :
A 50 meter long train passes over a bridge at the speed of 30 km per hour. If it takes 36 seconds to cross the bridge, what is the length of the bridge?
A train takes 5 minutes to cross a telegraphic post. Then the time taken by another train whose length is just double of the first train and moving with same speed to cross a platform of its own length is :
A train speeds past a pole in 20 seconds and speeds past a platform 100 meters in length in 30 seconds. What is the length of the train?
The time taken by a train 180 m long, travelling at 42 kmph, in passing a person walking in the same direction at 6 kmph, will be
Two trains 200 meters and 150 meters long are running on parallel rails in the same direction at speed of 40 km/hr and 45 km/hr respectively. Time taken by the faster train to cross the slowed train will be:
A train with 90 km/hr crosses a bridge in 36 seconds. Another train 100 meters shorter crosses the same bridge at 45 km/hr. What is the time taken by the second train to cross the bridge?
A train 125 m long passes a man, running at 5 kmph in the same direction in which the train is going, in 10 seconds. The speed of the train is:
Two, trains, one from Howrah to Patna and the other from Patna to Howrah, start simultaneously. After they meet, the trains reach their destinations after 9 hours and 16 hours respectively. The ratio of their speeds is:
A 100 m long train is going at a speed of 60 km/hr. It will cross a 140 m long railway bridge in-
A train 132 m long passes a telegraph pole in 6 seconds. Find the speed of the train?
A train running at the speed of 60 kmph crosses a 200 m long platform in 27 seconds. What is the length of the train?
A train running at a speed of 90 km/hr crosses a platform double its length in 36 seconds. What is the length of the platform in meters?
A train of length 150 meters takes 40.5 seconds to cross a tunnel of length 300 meters. What is the speed of the train in km/hr?
A 280 meter long train crosses a platform thrice its length in 50 seconds. What is the speed of the train in km/hr?
A train 110 meters long is running with a speed of 60 kmph. In what time will it pass a man who is running at 6 kmph in the direction opposite to that in which the train is going?
Two trains A and B start running together from the same point in the same direction, at the speed of 60 kmph and 72 kmph respectively. If the length of each of the trains is 240 meters, how long will it take for B to cross train A?
Two trains are moving in opposite directions @60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. The time taken by the slower train to cross the faster train in second is?
Two trains of lenths 120 m and 90 m are running with speed of 80 km/hr and 55 km/hr respectively towards each other on parallel lines. If they are 90 m apart, after how many seconds they will cross each other?
Two trains are coming from opposite directions with speed of 75 km/hr and 100 km/hr on to parallel tracks. At some moment the distance between them is 100km. After T hours, distance between them is again 100 km. T is equal to?
A train, 240 m long, crosses a man walking alone the line in opposite direction at the rate of 3 kmph in 10 seconds. The speed of the train is?
Two trains of equal length are running on parallel lines in the same directions at 46 km/hr and 36 km/hr. The faster train passes the slower train in 36 seconds. The length of each train is?
Two trains of equal lengths takes 10 seconds and 15 seconds respectively to cross a telegraph post. If the length of each train be 120 miters, in what time ( in seconds) will they cross each other traveling in opposite direction?
A train B speeding with 120 kmph crosses another train C running in the same direction, in 2 minutes. If the lengths of the trains B and C be 100m and 200m respectively, what is the speed (in kmph) of the train C?
What is the speed of a train if it overtakes two persons who are walking in the same direction at the rate of a m/s and (a + 1) m/s and passes them completely in b seconds and (b + 1) seconds respectively?
A train passes a 50 meter long platform in 14 seconds and a man standing on platform 10 seconds.The speed of the train is?
A train is moving at a speed of 132 km/hr. If the length of the train is 110 meters, how long it will take to cross a railway platform 165 meter long?
A train of length 500 feet crosses a platform of length 700 feet in 10 seconds. The speed of the train is?
Two trains start simultaneously (with uniform speeds) from two stations 270 km apart, each to the opposite station
they reach their destinations in $$6frac{1}{4}$$ hours and 4 hours after they meet. The rate at which the slower train travels is :
Answer Key
& { ext{At the time of meeting ,}} cr
& { ext{let the distance travelled by the}} cr
& { ext{first train be }}x{ ext{ km}}{ ext{.}} cr
& { ext{Then distance travelled by the }} cr
& { ext{second train is (}}x{ ext{ + 60) km}} cr
& herefore frac{x}{{16}} = frac{{x + 60}}{{21}} cr
& Rightarrow 21x = 16x + 960 cr
& Rightarrow 5x = 960 Rightarrow x = 192 cr
& { ext{Hence,}} cr
& { ext{distance between two stations}} cr
& { ext{ = (192 + 192 + 60) km}} cr
& { ext{ = 444 km}}{ ext{.}} cr} $$
& { ext{In these type of questions use the given}} cr
& { ext{below formula to save your valuable time}} cr
& frac{{{{ ext{S}}_1}}}{{{{ ext{S}}_2}}}{ ext{ = }}sqrt {frac{{{{ ext{T}}_2}}}{{{{ ext{T}}_1}}}} { ext{ }} cr
& { ext{Where }}{{ ext{S}}_1}{ ext{,}}{{ ext{S}}_2}{ ext{ and }}{{ ext{T}}_1}{ ext{, }}{{ ext{T}}_2}{ ext{ are the respective}} cr
& { ext{speeds and times of the objects}} cr
& Rightarrow frac{{45}}{{{{ ext{S}}_2}}} = sqrt {3frac{1}{3} div 4frac{4}{5}} cr
& { ext{ = }}{{ ext{S}}_2}{ ext{ = 45}} imes frac{6}{5}{ ext{ = 54 km/hr}} cr
& herefore { ext{Required speed = 54 km/hr}} cr} $$
& = frac{{390}}{{28 - 7}} imes 7 cr
& = frac{{390}}{{21}} imes 7 cr
& = frac{{390}}{3} cr
& = 130,{ ext{m}} cr} $$
& { ext{Length of the train}} cr
& { ext{ = Speed }} imes { ext{time}} cr
& { ext{ = 36 km/hr}} imes { ext{10 sec}} cr
& { ext{ = 36}} imes frac{5}{{18}}{ ext{m/s}} imes 10sec cr
& = 100{ ext{ metres}} cr
& { ext{Therefore, }} cr
& { ext{Time taken by train to cross a plateform}} cr
& { ext{ of 55 metre long in time}} cr
& { ext{ = }}frac{{left( {100 + 55}
ight)}}{{36 imes frac{5}{{18}}}} cr
& = frac{{155}}{{10}} cr
& { ext{Time}} = 15frac{1}{2},sec cr} $$
& { ext{Relative speed}} cr
& { ext{ = 60 + 50}} cr
& { ext{ = 110 km/h}} cr
& { ext{Time taken}} cr
& { ext{ = }}frac{{220}}{{110}} cr
& { ext{ = 2 hr}} cr} $$
& { ext{Relative speed of man & train}} cr
& { ext{ = }}frac{{100 imes 5}}{{36}} imes frac{{18}}{5} cr
& { ext{ = 50km/hr}} cr
& herefore { ext{speed of train}} cr
& { ext{ = 50}} - { ext{5}} cr
& { ext{ = 45 km/hr}} cr} $$
& { ext{Time taken by train to cross a pole}} cr
& { ext{ = 9 sec}} cr
& { ext{Distance covered in crossing a pole}} cr
& { ext{ = length of train}} cr
& { ext{Speed of the train}} cr
& { ext{ = 48 km/h}} cr
& = left( {frac{{48 imes 5}}{{18}}}
ight)m/sec cr
& = frac{{40}}{3}m/sec cr
& herefore { ext{Length of the train}} cr
& { ext{ = Speed }} imes { ext{Time}} cr
& { ext{ = }}frac{{40}}{3} imes 9 cr
& { ext{ = 120 m}} cr} $$
& { ext{Let the trains meet after t hours}} cr
& { ext{Speed of train A}} cr
& { ext{ = 75 km/hr}} cr
& { ext{Speed of train B}} cr
& { ext{ = 50 km/hr}} cr
& { ext{Distance covered by train A}} cr
& { ext{ = 75}} imes { ext{t = 75t}} cr
& { ext{Distance covered by train B}} cr
& { ext{ = 50}} imes { ext{t = 50t}} cr
& { ext{Distance}},{ ext{ = Speed }} imes { ext{Time}} cr
& { ext{According to question}} cr
& 75{ ext{t}} - 50{ ext{t}} = 175 cr
& Rightarrow 25{ ext{t}} = 175 cr
& Rightarrow { ext{t}} = frac{{175}}{{25}} = 7,{ ext{hour}} cr
& herefore { ext{Distance between A and B }} cr
& { ext{ = 75t}} + 50{ ext{t}} = 125{ ext{t}} cr
& = 125 imes 7 = 875,{ ext{km}} cr} $$
& { ext{Time taken by trains to cross each }} cr
& { ext{other in opposite direction}} cr
& { ext{ = }}frac{{{l_1} + {l_2}}}{{{ ext{relative speed in opposite direction}}}} cr
& { ext{ = }}frac{{left( {180 + 120}
ight)}}{{left( {65 + 55}
ight)}} cr
& { ext{ = }}frac{{300}}{{120 imes frac{5}{{18}}}} cr
& { ext{ = 9 seconds}} cr} $$
& { ext{Let the speed of first train be }} cr
& {{ ext{S}}_1}{ ext{ km/hr and speed of second train}} cr
& { ext{is }}{{ ext{S}}_2}{ ext{km/hr }} cr
& { ext{As we know,}} cr
& { ext{Time }} cr
& { ext{ = }}frac{{{ ext{total distance}}}}{{{ ext{relative speed in same/opposite direction}}}} cr
& { ext{In the same direction}} cr
& Rightarrow { ext{27 sec = }}frac{{left( {100 + 95}
ight)}}{{left( {{ ext{ }}{{ ext{S}}_1} - { ext{ }}{{ ext{S}}_2}}
ight) imes frac{5}{{18}}}} cr
& Rightarrow 27 = frac{{195 imes 18}}{{left( {{ ext{ }}{{ ext{S}}_1} - { ext{ }}{{ ext{S}}_2}}
ight) imes 5}} cr
& Rightarrow { ext{ }}{{ ext{S}}_1} - { ext{ }}{{ ext{S}}_2} = 26.......................(i) cr
& { ext{In the opposite direction,}} cr
& Rightarrow 9 = frac{{left( {100 + 95}
ight)}}{{left( {{ ext{ }}{{ ext{S}}_1}{ ext{ + }}{{ ext{S}}_2}}
ight) imes frac{5}{{18}}}} cr
& Rightarrow 9 = frac{{195 imes 18}}{{left( {{ ext{ }}{{ ext{S}}_1}{ ext{ + }}{{ ext{S}}_2}}
ight) imes 5}} cr
& Rightarrow { ext{ }}{{ ext{S}}_1}{ ext{ + }}{{ ext{S}}_2} = 39 imes 2 cr
& Rightarrow { ext{ }}{{ ext{S}}_1}{ ext{ + }}{{ ext{S}}_2} = 78 cr
& { ext{From equation (i) and (ii)}} cr
& Rightarrow { ext{ }}{{ ext{S}}_1} - { ext{ }}{{ ext{S}}_2} = 26 cr
& Rightarrow { ext{ }}{{ ext{S}}_1}{ ext{ + }}{{ ext{S}}_2} = 78 cr
& Rightarrow { ext{ }}{{ ext{S}}_1} = frac{{26 + 78}}{2} cr
& Rightarrow { ext{ }}{{ ext{S}}_1} = frac{{104}}{2} cr
& Rightarrow { ext{ }}{{ ext{S}}_1}{ ext{ = 52 km/hr and }} cr
& ,,,,,,,,,,{{ ext{S}}_2}{ ext{ = 26 km/hr}} cr} $$
& { ext{Let length of train }} cr
& { ext{ = }}l,{ ext{metre}} cr
& Rightarrow { ext{Time }} cr
& { ext{ = }}frac{{{ ext{total distance}}}}{{{ ext{relative speed in opposite direction}}}} cr
& Rightarrow 4sec , = ,frac{{l + 0}}{{left( {84 + 6}
ight) imes frac{5}{{18}}{ ext{m/s}}}} cr
& Rightarrow 4, = frac{l}{{90 imes frac{5}{{18}}}} cr
& Rightarrow ,l, = ,100,{ ext{m}} cr
& herefore { ext{ length of the train = 100 m}} cr} $$
& { ext{Let the length of train }}x{ ext{ m }} cr
& { ext{Speed of train }} cr
& { ext{ = }}frac{{left( {{ ext{Length of train + length of bridge }}}
ight)}}{{{ ext{Time taken in crossing}}}}{ ext{ }} cr
& { ext{According to information we get}} cr
& Rightarrow frac{{x + 500}}{{100}} = frac{{x + 250}}{{60}} cr
& Rightarrow 60left( {x + 500}
ight) = 100left( {x + 250}
ight) cr
& Rightarrow 3left( {x + 500}
ight) = 5left( {x + 250}
ight) cr
& Rightarrow 5x + 1250 = 3x + 1500 cr
& Rightarrow 5x - 3x = 1500 - 1250 cr
& Rightarrow 2x = 250 cr
& Rightarrow x = frac{{250}}{2} = 125,{ ext{m}} cr} $$
& { ext{Let the length of train be x m}} cr
& { ext{When a train crosses a light }} cr
& { ext{post in 3 second the distance covered}} cr
& { ext{ = length of train }} cr
& Rightarrow { ext{speed of train = }}frac{x}{3} cr
& { ext{Distance covered in crossing a}} cr
& { ext{900 meter platfrom in 30 seconds}} cr
& { ext{ = Length of platfrom + length of train}} cr
& { ext{Speed of train = }}frac{{x + 900}}{30} cr
& Rightarrow frac{x}{3} = frac{{x + 900}}{{30}}left[ {x08ecause { ext{Speed = }}frac{{{ ext{Distance}}}}{{{ ext{Time}}}}}
ight] cr
& Rightarrow frac{x}{1} = frac{{x + 900}}{{10}} cr
& Rightarrow 10x = x + 900 cr
& Rightarrow 10x - x = 900 cr
& Rightarrow 9x = 900 cr
& Rightarrow x = frac{{900}}{9} = 100{ ext{m}} cr
& { ext{When the length of the platform be 800m,}} cr
& { ext{then time T be taken by train to cross 800m}} cr
& { ext{long platfrom}} cr
& frac{x}{3} = frac{{x + 800}}{T} cr
& Rightarrow Tx = 3x + 2400 cr
& Rightarrow 100T = 300 + 2400 cr
& Rightarrow 100T = 2700 cr
& Rightarrow T = frac{{2700}}{{100}} = 27{ ext{ seconds}} cr} $$
& { ext{Given , }} cr
& { ext{Train cover 3584 kms in 2 days 8 hours}} cr
& left( {2,{ ext{days 8 hours = }}frac{7}{3}{ ext{ days}}}
ight) cr
& { ext{Average speed = }} {frac{{3584}}{{ {frac{{7}}{{3}}}}}} cr
& { ext{ = 1536 km/day = }}frac{{1536}}{{24}}{ ext{ = 64 km/h}} cr
& { ext{Distance covered in two days}} cr
& { ext{ = 1440 + 1608 = 3048 km}} cr
& { ext{Remaining distance for third day}} cr
& { ext{ = 3584 }} - { ext{3048 = 536 km}} cr
& { ext{Third day 536 km is covered in }} cr
& { ext{8 hour with speed of}} cr
& { ext{ = }}frac{{536}}{8} = 67{ ext{ km/h }} cr
& { ext{( 3rd day total 536 km distance}} cr
& { ext{ covered by 67 km/hr in 8 hr)}} cr
& herefore { ext{Difference of average speedm}} cr
& { ext{ = 67}} - { ext{64 = 3 km/hr}} cr} $$
& { ext{Let the length of train be x m}} cr
& { ext{When a train crosses a light }} cr
& { ext{post in 9 second the distance covered}} cr
& { ext{ = length of train }} cr
& Rightarrow { ext{speed of train = }}frac{x}{9} cr
& { ext{Distance covered in crossing a}} cr
& { ext{700 meter platfrom in 30 seconds}} cr
& { ext{ = Length of platfrom + length of train}} cr
& { ext{Speed of train = }}frac{{x + 700}}{30} cr
& Rightarrow frac{x}{9} = frac{{x + 700}}{{30}}left[ {x08ecause { ext{Speed = }}frac{{{ ext{Distance}}}}{{{ ext{Time}}}}}
ight] cr
& Rightarrow frac{x}{3} = frac{{x + 700}}{{10}} cr
& Rightarrow 10x = 3x + 2100 cr
& Rightarrow 10x - 3x = 2100 cr
& Rightarrow 7x = 2100 cr
& Rightarrow x = frac{{2100}}{7} = 300{ ext{m}} cr
& { ext{When the length of the platform be 800m,}} cr
& { ext{then time T be taken by train to cross 800m}} cr
& { ext{long platform}} cr
& frac{x}{9} = frac{{x + 800}}{T} cr
& Rightarrow Tx = 9x + 7200 cr
& Rightarrow 300T = 2700 + 7200 cr
& Rightarrow 300T = 9900 cr
& Rightarrow T = frac{{9900}}{{300}} = 33{ ext{ seconds}} cr} $$
& { ext{Speed of train A}} cr
& { ext{ = 63 kmph}} cr
& { ext{ = }}left( {frac{{63 imes 5}}{{18}}}
ight){ ext{m/sec}} cr
& { ext{ = 17}}{ ext{.5 m/sec}} cr
& { ext{Speed of train B}} cr
& { ext{ = 54 kmph}} cr
& { ext{ = }}left( {frac{{54 imes 5}}{{18}}}
ight){ ext{m/sec = 15 m/sec}} cr
& { ext{If the length of train A be }}x{ ext{ metre,}} cr
& { ext{then}} cr
& { ext{Speed of train A}} cr
& { ext{ = }}frac{{{ ext{Length of train + length of platform}}}}{{{ ext{Time taken in crossing}}}}{ ext{ }} cr
& Rightarrow 17.5 = frac{{x + 199.5}}{{21}} cr
& Rightarrow 17.5 imes 21 = x + 199.5 cr
& Rightarrow 367.5 = x + 199.5 cr
& Rightarrow x = 367.5 - 199.5 cr
& Rightarrow 168,{ ext{metres}} cr
& { ext{Relative speed}} cr
& { ext{ = ( Speed train A + Speed train B)}} cr
& { ext{ = (17}}{ ext{.5 + 15) m/sec}} cr
& { ext{ = 32}}{ ext{.5 m/sec}} cr
& { ext{Required time}} cr
& { ext{ = }}frac{{{ ext{ Length of train A + Length of train B}}}}{{{ ext{Relative speed }}}} cr
& = left( {frac{{168 + 157}}{{32.5}}}
ight){ ext{seconds}} cr
& = 10,{ ext{seconds}} cr} $$
& { ext{Average speed of train}} cr
& { ext{ = 40 km/hr}} cr
& { ext{Reach at its destination at on time }} cr
& { ext{New average speed of train}} cr
& { ext{ = 35 km/h}} cr
& { ext{Time = 15 minutes}} cr
& ,,,,,,,,,,,,,,,{ ext{ = }}frac{{15}}{{60}}{ ext{hours }} cr
& { ext{Then distance travelled}} cr
& { ext{ = }}frac{{40 imes 35}}{{40 - 35}}{ ext{ }} imes frac{{15}}{{60}} cr
& { ext{ = }}frac{{40 imes 35}}{5}{ ext{ }} imes frac{{15}}{{60}} cr
& { ext{ = 70}},{ ext{km}} cr} $$
& { ext{Distance = Speed }} imes { ext{Time}} cr
& { ext{Distance covered by train with the}} cr
& { ext{speed of 30 kmph in 12 minutes is }} cr
& { ext{ = 30}} imes frac{{12}}{{60}} = 6{ ext{km}} cr
& { ext{Distance covered by the same train}} cr
& { ext{with the speed of 45 kmph in 8 minutes is }} cr
& { ext{ = 45}} imes frac{8}{{60}} = 6{ ext{km}} cr
& { ext{Average speed}} cr
& { ext{ = }}frac{{{ ext{total distance}}}}{{{ ext{total time}}}}. cr
& Rightarrow frac{{left( {6 + 6}
ight){ ext{km}}}}{{left( {12 + 8}
ight)min }} = frac{{12}}{{20}} imes 60 cr
& { ext{ = 36 kmph}} cr} $$
& { ext{Speed}} = left( {60 imes frac{5}{{18}}}
ight){ ext{m/sec}} = {frac{{50}}{3}} { ext{m/sec}} cr
& { ext{Length}},{ ext{of}},{ ext{the}},{ ext{train}} = left( {{ ext{Speed}} imes { ext{Time}}}
ight) cr
& herefore { ext{Length}},{ ext{of}},{ ext{the}},{ ext{train}} cr
& = left( {frac{{50}}{3} imes 9}
ight)m = 150m cr} $$
& { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{relative}},{ ext{to}},{ ext{man}} cr
& = {frac{{125}}{{10}}} { ext{ m/sec}} cr
& = {frac{{25}}{2}} { ext{ m/sec}} cr
& = {frac{{25}}{2} imes frac{{18}}{5}} { ext{ km/hr}} cr
& = 45,{ ext{km/hr}} cr
& { ext{Let}},{ ext{the}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{km/hr}}. cr
& ext{Then, relative speed} = left( {x - 5}
ight),{ ext{km/hr}} cr
& herefore x - 5 = 45 cr
& Rightarrow x = 50,{ ext{km/hr}} cr} $$
& { ext{Speed}} = {45 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,, = {frac{{25}}{2}} ,{ ext{m/sec}} cr
& { ext{Time}} = 30,{ ext{sec}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{bridge}},{ ext{be}},x,{ ext{metres}} cr
& { ext{Then}},,frac{{130 + x}}{{30}} = frac{{25}}{2} cr
& Rightarrow 2left( {130 + x}
ight) = 750 cr
& Rightarrow x = 245,m cr} $$
& { ext{Let}},{ ext{the}},{ ext{speeds}},{ ext{of}},{ ext{the}},{ ext{two}},{ ext{trains}},{ ext{be}},x,{ ext{m/sec}} cr
& { ext{and}},y,{ ext{m/sec}},{ ext{respectively}}. cr
& { ext{Then,}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{first}},{ ext{train}} = 27x,{ ext{metres}}, cr
& { ext{and}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{second}},{ ext{train}} = 17y,{ ext{metres}}. cr
& herefore frac{{27x + 17y}}{{x + y}} = 23 cr
& Rightarrow 27x + 17y = 23x + 23y cr
& Rightarrow 4x = 6y cr
& Rightarrow frac{x}{y} = frac{3}{2} cr} $$
& { ext{Speed}} = {54 imes frac{5}{{18}}} ,{ ext{m/sec}} = 15,{ ext{m/sec}} cr
& { ext{Length}},{ ext{of}},{ ext{the}},{ ext{train}} = left( {15 imes 20}
ight){ ext{m}} = 300,{ ext{m}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{platform}},{ ext{be}},x,{ ext{metres}} cr
& { ext{Then}},,frac{{x + 300}}{{36}} = 15 cr
& Rightarrow x + 300 = 540 cr
& Rightarrow x = 240,{ ext{m}} cr} $$
& { ext{Speed}} = {frac{{240}}{{24}}} ,{ ext{m/sec}} = 10,{ ext{m/sec}} cr
& herefore { ext{Required}},{ ext{time}} cr
& { ext{ = }}, {frac{{240 + 650}}{{10}}} ,{ ext{sec}}. cr
& = 89,sec. cr} $$
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{each}},{ ext{train}},{ ext{be}},x,{ ext{metres}}. cr
& { ext{Then,}},{ ext{distance}},{ ext{covered}} = 2x,{ ext{metres}}. cr
& { ext{Relative}},{ ext{speed}} cr
& = left( {46 - 36}
ight),{ ext{km/hr}} cr
& = {10 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = {frac{{25}}{9}} ,{ ext{m/sec}} cr
& herefore frac{{2x}}{{36}} = frac{{25}}{9} cr
& Rightarrow 2x = 100 cr
& Rightarrow x = 50 cr} $$
& { ext{Formula}},{ ext{for}},{ ext{converting}},{ ext{from}},{ ext{km/hr}},{ ext{to}},{ ext{m/s:}} cr
& X,{ ext{km/hr}} = {X imes frac{5}{{18}}} ,{ ext{m/s}} cr
& { ext{Therefore,}},{ ext{Speed}} cr
& = {45 imes frac{5}{{18}}} ,{ ext{m/sec}} = frac{{25}}{2}{ ext{m/sec}} cr
& { ext{Total}},{ ext{distance}},{ ext{to}},{ ext{be}},{ ext{covered}} cr
& = left( {360 + 140}
ight)m = 500,m cr
& { ext{Formula}},{ ext{for}},{ ext{finding}},{ ext{Time}} cr
& = {frac{{{ ext{Distance}}}}{{{ ext{Speed}}}}} cr
& herefore { ext{Required}},{ ext{time}} cr
& = left( {frac{{500 imes 2}}{{25}}}
ight),sec cr
& = 40,sec . cr} $$
& { ext{Relative}},{ ext{speed}} = left( {60 + 90}
ight),{ ext{km/hr}} cr
& = {150 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = {frac{{125}}{3}} ,{ ext{m/sec}} cr
& { ext{Distance}},{ ext{covered}} cr
& = left( {1.10 + 0.9}
ight),km cr
& = 2,km cr
& = ,2000,m cr
& { ext{Required}},{ ext{time}} cr
& = {2000 imes frac{3}{{125}}} ,{ ext{sec}} cr
& = 48,sec cr} $$
& { ext{Speed}},{ ext{of}},{ ext{train}},{ ext{relative}},{ ext{to}},{ ext{jogger}} cr
& = left( {45 - 9}
ight),{ ext{km/hr}} cr
& = 36,{ ext{km/hr}} cr
& {36 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = 10,{ ext{m/sec}} cr
& { ext{Distance}},{ ext{to}},{ ext{be}},{ ext{covered}} cr
& = left( {240 + 120}
ight),m cr
& = 360,m cr
& herefore { ext{Time}},{ ext{taken}} cr
& = {frac{{360}}{{10}}} ,{ ext{sec}} cr
& = 36,{ ext{sec}} cr} $$
& { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{relative}},{ ext{to}},{ ext{man}} cr
& = left( {63 - 3}
ight),{ ext{km/hr}} cr
& = 60,{ ext{km/hr}} cr
& = left( {60 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& = frac{50}{3}, ext{m/sec} cr
& herefore { ext{Time}},{ ext{taken}},{ ext{to}},{ ext{pass}},{ ext{the}},{ ext{man}} cr
& = left( {500 imes frac{3}{{50}}}
ight),sec cr
& = 30,sec cr} $$
& { ext{Relative}},{ ext{speed}} cr
& = left( {45 + 30}
ight),{ ext{km/hr}} cr
& = left(75 imes frac{5}{18}
ight), ext{m/sec} cr
& = {frac{{125}}{6}} ,{ ext{m/sec}} cr
& { ext{We}},{ ext{have}},{ ext{to}},{ ext{find}},{ ext{the}},{ ext{time}},{ ext{taken}},{ ext{by}},{ ext{the}} cr
& { ext{slower}},{ ext{train}},{ ext{to}},{ ext{pass}},{ ext{the}},{ ext{DRIVER}},{ ext{of}}, cr
& { ext{The}},{ ext{faster}},{ ext{train}},{ ext{and}},{ ext{not}},{ ext{the}},{ ext{complete}},{ ext{train}}{ ext{.}} cr
& cr
& { ext{So,}},{ ext{distance}},{ ext{covered = Length}},{ ext{of}},{ ext{the}},{ ext{slower}},{ ext{train}}. cr
& { ext{Therefore,}},{ ext{Distance}},{ ext{covered = 500}},{ ext{m}}. cr
& herefore { ext{Required}},{ ext{time}} cr
& = {500 imes frac{6}{{125}}} cr
& = 24,sec cr} $$
& { ext{Let}},{ ext{the}},{ ext{speed}},{ ext{of}},{ ext{each}},{ ext{train}},{ ext{be}},x,{ ext{m/sec}}. cr
& { ext{Then,}},{ ext{relative}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{two}},{ ext{trains}} = 2x,{ ext{m/sec}} cr
& { ext{So}},,2x = frac{{ {120 + 120} }}{{12}} cr
& Rightarrow 2x = 20 cr
& Rightarrow x = 10 cr
& herefore { ext{Speed}},{ ext{of}},{ ext{each}},{ ext{train}} = 10,{ ext{m/sec}} cr
& = {10 imes frac{{18}}{5}} ,{ ext{km/hr}} cr
& = 36,{ ext{km/hr}} cr} $$
& { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{first}},{ ext{train}} cr
& = {frac{{120}}{{10}}} ,{ ext{m/sec}} cr
& = 12,{ ext{m/sec}} cr
& { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{second}},{ ext{train}} cr
& {frac{{120}}{{15}}} ,{ ext{m/sec}} cr
& = 8,{ ext{m/sec}} cr
& { ext{Relative}},{ ext{speed}} = {12 + 8} = 20,{ ext{m/sec}} cr
& herefore { ext{Required}},{ ext{time}} cr
& = {frac{{ {120 + 120} }}{{20}}} ,{ ext{ sec}} cr
& = 12,{ ext{sec}} cr} $$
& { ext{Let}},{ ext{the}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{second}},{ ext{train}},{ ext{be}},x,{ ext{km/hr}}. cr
& { ext{Relative}},{ ext{speed}}, cr
& = ,left( {x + 50}
ight),{ ext{km/hr}} cr
& = left[ {left( {x + 50}
ight) imes frac{5}{{18}}}
ight],{ ext{m/sec}} cr
& = {frac{{250 + 5x}}{{18}}} ,{ ext{m/sec}} cr
& { ext{Distance}},{ ext{covered}} cr
& = left( {108 + 112}
ight) = 220,m cr
& herefore frac{{220}}{{ {frac{{250 + 5x}}{{18}}} }} = 6 cr
& Rightarrow 250 + 5x = 660 cr
& Rightarrow x = 82,{ ext{km/hr}} cr} $$
& { ext{Relative}},{ ext{speed}} = left( {40 - 20}
ight),{ ext{km/hr}} cr
& = left( {20 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& = {frac{{50}}{9}} ,{ ext{m/sec}} cr
& herefore { ext{Length}},{ ext{of}},{ ext{faster}},{ ext{train}} cr
& = left( {frac{{50}}{9} imes 5}
ight),m cr
& = frac{{250}}{9},m cr
& = 27frac{7}{9},m cr} $$
& 2,kmph = left( {2 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,, = frac{5}{9},{ ext{m/sec}} cr
& 4,kmph = left( {4 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,, = frac{{10}}{9},{ ext{m/sec}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{metres}}, cr
& { ext{and}},{ ext{its}},{ ext{speed}},{ ext{be}},y,{ ext{m/sec}} cr
& { ext{Then}},, {frac{x}{{y - frac{5}{9}}}} = 9,{ ext{and}}, {frac{x}{{y - frac{{10}}{9}}}} = 10 cr
& herefore 9y - 5 = x,{ ext{and}},10left( {9y - 10}
ight) = 9x cr
& Rightarrow 9y - x = 5,{ ext{and}},90y - 9x = 100 cr
& { ext{On}},{ ext{solving,}},{ ext{we}},{ ext{get}}:,x = 50 cr
& herefore { ext{Length}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{is}},50,m cr} $$
& 4.5,{ ext{km/hr}} = left( {4.5 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = frac{5}{4},{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 1.25,{ ext{m/sec,}},{ ext{and}} cr
& 5.4,km/hr = left( {5.4 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = frac{3}{2},{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 1.5,{ ext{m/sec}} cr
& { ext{Let}},{ ext{the}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{m/sec}} cr
& { ext{Then}},,left( {x - 1.25}
ight) imes 8.4 = left( {x - 1.5}
ight) imes 8.5 cr
& Rightarrow 8.4x - 10.5 = 8.5x - 12.75 cr
& Rightarrow 0.1x = 2.25 cr
& Rightarrow x = 22.5 cr
& herefore { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{train}} cr
& = left( {22.5 imes frac{{18}}{5}}
ight),{ ext{km/hr}} cr
& = 81,{ ext{km/hr}} cr} $$
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{first}},{ ext{train}},{ ext{be}},x,{ ext{metres}} cr
& { ext{Then,}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{second}},{ ext{train}},{ ext{is}}, {frac{x}{2}} ,{ ext{metres}} cr
& { ext{Relative}},{ ext{speed}} = left( {48 + 42}
ight),{ ext{kmph}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = left( {90 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 25,{ ext{m/sec}} cr
& herefore frac{{ {x + left( {x/2}
ight)} }}{{25}} = 12 cr
& or,frac{{3x}}{2} = 300 cr
& or,x = 200 cr
& herefore { ext{Length}},{ ext{of}},{ ext{first}},{ ext{train}} = 200,{ ext{m}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{platform}},{ ext{be}},y,{ ext{metres}} cr
& { ext{Speed}},{ ext{of}},{ ext{the}},{ ext{first}},{ ext{train}} cr
& = left( {48 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& = frac{{40}}{3},{ ext{m/sec}} cr
& herefore left( {200 + y}
ight) imes frac{3}{{40}} = 45 cr
& Rightarrow 600 + 3y = 1800 cr
& Rightarrow y = 400,{ ext{m}} cr} $$
& { ext{Suppose}},{ ext{they}},{ ext{meet}},x,{ ext{hours}},{ ext{after}},{ ext{7}},{ ext{a}}{ ext{.m}}. cr
& { ext{Distance}},{ ext{covered}},{ ext{by}},{ ext{A}}, cr
& { ext{in}},x,{ ext{hours = 20x}},{ ext{km}}{ ext{.}} cr
& { ext{Distance}},{ ext{covered}},{ ext{by}},{ ext{B}} cr
& ,{ ext{in}},left( {x - 1}
ight),{ ext{hours}} = 25left( {x - 1}
ight),km cr
& herefore 20x + 25left( {x - 1}
ight) = 110 cr
& Rightarrow 45x = 135 cr
& Rightarrow x = 3 cr
& { ext{So,}},{ ext{they}},{ ext{meet}},{ ext{at}},{ ext{10}},{ ext{a}}{ ext{.m}}{ ext{.}}, cr} $$
& = frac{{75}}{{7.5}}{ ext{m/sec}} = 10,{ ext{m/sec}} cr
& = left( {10 imes frac{{18}}{5}}
ight){ ext{km/hr}} = 36,{ ext{km/hr}} cr} $$ Let the speed of the train be x km/hr. Then, relative speed = (x - 6) km/hr ∴ x - 6 = 36 ⇒ x = 42 km/hr Speed of the train relative to second man $$eqalign{
& { ext{ = }}frac{{75}}{{6frac{3}{4}}},{ ext{m/sec}} cr
& = left( {75 imes frac{4}{{27}}}
ight){ ext{m/sec}} cr
& = frac{{100}}{9}{ ext{m/sec}} cr
& = left( {frac{{100}}{9} imes frac{{18}}{5}}
ight){ ext{km}} cr
& = 40,{ ext{km/hr}} cr} $$ Let the speed of the second man be y kmph. Then, relative speed = (42 - y) kmph ∴ 42 - y = 40 ⇒ y = 2 km/hr
ight)$$ xa0km/hr = 36 km/hr
ight)$$ xa0kmph = 54 kmph
& = left[ {frac{{{ ext{x}} + { ext{x}}}}{{left( {frac{{ ext{x}}}{3} + frac{{ ext{x}}}{4}}
ight)}}}
ight]{ ext{sec}} cr
& = left[ {frac{{2{ ext{x}}}}{{left( {frac{{7{ ext{x}}}}{{12}}}
ight)}}}
ight]{ ext{sec}} cr
& = left( {2 imes frac{{12}}{7}}
ight){ ext{sec}} cr
& = frac{{24}}{7}{ ext{sec}} cr
& = 3frac{3}{7}{ ext{sec}} cr} $$
ight){ ext{m}} = left( {frac{{4{ ext{x}}}}{3}}
ight){ ext{m}}$$ ∴ Time taken by trains to cross each other $$eqalign{
& = left( {frac{{{ ext{x}} + frac{{4{ ext{x}}}}{3}}}{{2{ ext{y}}}}}
ight){ ext{sec}} cr
& = frac{{7{ ext{x}}}}{{6{ ext{y}}}} cr
& = frac{7}{6} imes frac{{ ext{x}}}{{ ext{y}}} cr
& = left( {frac{7}{6} imes 120}
ight){ ext{sec}} cr
& = 140,{ ext{sec}} cr} $$ Hence, difference in times taken = (140 - 120) sec = 20 sec
& { ext{Relative}},{ ext{speed}} cr
& = left( {120 + 80}
ight),{ ext{km/hr}} cr
& = {200 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = {frac{{500}}{9}} ,{ ext{m/sec}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{other}},{ ext{train}},{ ext{be}},{ ext{x}},{ ext{metres}}{ ext{.}} cr
& { ext{Then,}},frac{{x + 270}}{9} = frac{{500}}{9} cr
& Rightarrow x + 270 = 500 cr
& Rightarrow x = 230 cr} $$
& { ext{Speed}} = {72 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,, = 20,{ ext{m/sec}} cr
& { ext{Time}} = 26,{ ext{sec}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{metres}}{ ext{.}} cr
& { ext{Then}},,frac{{x + 250}}{{26}} = 20 cr
& Rightarrow x + 250 = 520 cr
& Rightarrow x = 270 cr} $$
& { ext{Let}},{ ext{the}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{slower}},{ ext{train}},{ ext{be}},x,{ ext{m/sec}} cr
& { ext{Then,}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{faster}},{ ext{train}} = 2x,{ ext{m/sec}} cr
& { ext{Relative}},{ ext{speed}} = ,left( {x + 2x}
ight),{ ext{m/sec}} = 3x,{ ext{m/sec}} cr
& herefore frac{{ {100 + 100} }}{8} = 3x cr
& Rightarrow 24x = 200 cr
& Rightarrow x = frac{{25}}{3} cr
& { ext{So,}},{ ext{speed}},{ ext{of}},{ ext{the}},{ ext{faster}},{ ext{train}}, = frac{{50}}{3},{ ext{m/sec}} cr
& = {frac{{50}}{3} imes frac{{18}}{5}} ,{ ext{km/hr}} cr
& = 60,{ ext{km/hr}} cr} $$
& { ext{Relative}},{ ext{speed}} = left( {60 + 40}
ight),{ ext{km/hr}} cr
& = {100 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = {frac{{250}}{9}} ,{ ext{m/sec}}. cr
& { ext{Distance}},{ ext{covered}},{ ext{in}},{ ext{crossing}},{ ext{each}},{ ext{other}} cr
& = left( {140 + 160}
ight)m = 300,m cr
& { ext{Required}},{ ext{time}} cr
& = {300 imes frac{9}{{250}}} ,{ ext{sec}} cr
& = frac{{54}}{5},{ ext{sec}} cr
& = 10.8,{ ext{sec}} cr} $$
& { ext{Speed}},{ ext{of}},{ ext{train}},{ ext{relative}},{ ext{to}},{ ext{man}} cr
& = left( {60 + 6}
ight),{ ext{km/hr}} cr
& = 66,{ ext{km/hr}} cr
& = {66 imes frac{5}{{18}}} ,{ ext{m/sec}} cr
& = {frac{{55}}{3}} ,{ ext{m/sec}} cr
& herefore { ext{Time}},{ ext{taken}},{ ext{to}},{ ext{pass}},{ ext{the}},{ ext{man}} cr
& = {110 imes frac{3}{{55}}} { ext{sec}} = 6,{ ext{sec}} cr} $$
& { ext{Total}},{ ext{distance}},{ ext{covered}} cr
& = left( {frac{7}{2} + frac{1}{4}}
ight),{ ext{miles}} cr
& = frac{{15}}{4},{ ext{miles}} cr
& herefore { ext{Time}},{ ext{taken}} cr
& = left( {frac{{15}}{{4 imes 75}}}
ight),{ ext{hrs}} cr
& = frac{1}{{20}},{ ext{hrs}} cr
& = left( {frac{1}{{20}} imes 60}
ight),min cr
& = 3,min cr} $$
& { ext{Speed}} = left( {78 imes frac{5}{{18}}}
ight),{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,, = {frac{{65}}{3}} ,{ ext{m/sec}} cr
& { ext{Time = }},{ ext{1}},{ ext{minute = 60}},{ ext{second}}. cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{tunnel}},{ ext{be}},x,{ ext{metres}}. cr
& { ext{Then}},, {frac{{800 + x}}{{60}}} = frac{{65}}{3} cr
& Rightarrow 3left( {800 + x}
ight) = 3900 cr
& Rightarrow x = 500 cr} $$
& { ext{Speed}} = {frac{{300}}{{18}}} ,{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,, = frac{{50}}{3},{ ext{m/sec}} cr
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{platform}},{ ext{be}},x,{ ext{metres}}{ ext{.}} cr
& { ext{Then}}, {frac{{x + 300}}{{39}}} = frac{{50}}{3} cr
& Rightarrow 3left( {x + 300}
ight) = 1950 cr
& Rightarrow x = 350,m. cr} $$
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{metres}} cr
& ,{ ext{and}},{ ext{its}},{ ext{speed}},{ ext{by}},y,{ ext{m/sec}} cr
& Then,,frac{x}{y} = 15,,,,,, Rightarrow ,,,,,y = frac{x}{{15}} cr
& herefore frac{{x + 100}}{{25}} = frac{x}{{15}} cr
& Rightarrow 15left( {x + 100}
ight) = 25x cr
& Rightarrow 15x + 1500 = 25x cr
& Rightarrow 1500 = 10x cr
& Rightarrow x = 150m cr} $$
& { ext{Let}},{ ext{the}},{ ext{length}},{ ext{of}},{ ext{the}},{ ext{train}},{ ext{be}},x,{ ext{metres}} cr
& ,{ ext{and}},{ ext{its}},{ ext{speed}},{ ext{by}},y,{ ext{m/sec}} cr
& { ext{Then}},,frac{x}{y} = 8,,,,,, Rightarrow ,,,,,x = 8y cr
& { ext{Now}},,frac{{x + 264}}{{20}} = y cr
& Rightarrow 8y + 264 = 20y cr
& Rightarrow y = 22 cr
& herefore { ext{Speed}} = 22,{ ext{m/sec}} cr
& ,,,,,,,,,,,,,,,,,,,,,,, = {22 imes frac{{18}}{5}} ,{ ext{km/hr}} cr
& ,,,,,,,,,,,,,,,,,,,,,,, = 79.2,{ ext{km/hr}} cr} $$
ight)$$ = 3500 + 1 = 3501
ight)$$ xa0 m/sec = 25 m/sec Total distance covered = (120 + 230) m = 350 m ∴ Required time = $$frac{{350}}{{25}}$$ seconds = 14 seconds
ight)$$ xa0m/sec = $$frac{{25}}{3}$$ m/sec Time = 36 second Let the length of the bridge be x meters. Then, $$frac{{50 + { ext{x}}}}{{36}}$$ xa0 = $$frac{{25}}{3}$$ ⇒ 3(50 + x) = 900 ⇒ 50 + x = 300 ⇒ x = 250 meters
& = left[ {frac{{2{ ext{x}} + 2{ ext{x}}}}{{left( {frac{{ ext{x}}}{{300}}}
ight)}}}
ight]{ ext{sec}} cr
& = left( {frac{{4{ ext{x}} imes 300}}{{ ext{x}}}}
ight){ ext{sec}} cr
& = 1200,{ ext{sec}} cr
& = frac{{1200}}{{60}},{ ext{min}} cr
& = 20,{ ext{minutes}} cr} $$
ight)$$ xa0m/sec = 10 m/sec ∴ Time taken to pass the man = $$frac{{180}}{{10}}$$ sec = 18 sec
ight)$$ xa0m/sec = $$frac{{25}}{{18}}$$ m/sec Total distance covered = Sum of lengths of trains = (200 + 150) m = 350 m ∴ Time taken = $$left( {350 imes frac{{18}}{{25}}}
ight)$$ xa0 sec = 252 seconds
ight)$$ xa0m/sec = 25 m/sec Speed of the second train
= 45 km/hr = $$left( {45 imes frac{5}{{18}}}
ight)$$ xa0m/sec = $$frac{{25}}{2}$$ m/sec Then, $$frac{{{ ext{x}} + { ext{y}}}}{{36}}$$ = 25 ⇒ x + y = 900 ∴ Required time $$eqalign{
& = left[ {frac{{left( {{ ext{x}} - 100}
ight) + { ext{y}}}}{{frac{{25}}{2}}}}
ight]{ ext{sec}} cr
& = left[ {frac{{left( {{ ext{x}} + { ext{y}}}
ight) - 100}}{{frac{{25}}{2}}}}
ight]{ ext{sec}} cr
& = left( {800 imes frac{2}{{25}}}
ight){ ext{sec}} cr
& = 64,{ ext{sec}} cr} $$
& = frac{{125}}{{10}}{ ext{m/sec}} cr
& = frac{{25}}{2}{ ext{m/sec}} cr
& = left( {frac{{25}}{2} imes frac{{18}}{5}}
ight){ ext{m/sec}} cr
& = 45,{ ext{km/hr}} cr} $$ Let the speed of the train be x kmph. Then, relative speed = (x - 5) kmph ∴ x - 5 = 45 or x = 50 km/hr
& { ext{Let}},{ ext{us}},{ ext{name}},{ ext{the}},{ ext{trains}},{ ext{as}},{ ext{A}},{ ext{and}},{ ext{B}}{ ext{.}},{ ext{Then}}, cr
& left( {{ ext{A's}},{ ext{speed}}}
ight):left( {{ ext{B's}},{ ext{speed}}}
ight) cr
& = sqrt b :sqrt a cr
& = sqrt {16} :sqrt 9 cr
& = 4:3, cr} $$
& { ext{Speed }} cr
& { ext{ = }}left( {60 imes frac{5}{{18}}}
ight){ ext{m/sec}} cr
& { ext{ = }}frac{{50}}{3}{ ext{ m/sec}} cr
& { ext{Total distance covered}} cr
& { ext{ = (100 + 140) m = 240 m}} cr
& herefore { ext{Required time}} cr
& { ext{ = }}left( {240 imes frac{3}{{50}}}
ight){ ext{sec}} cr
& { ext{ = }}frac{{72}}{5}{ ext{sec}} cr
& { ext{ = 14}}{ ext{.4 sec}} cr} $$
& { ext{Speed}} cr
& { ext{ = }}left( {frac{{132}}{6}}
ight){ ext{m/sec}} cr
& { ext{ = }}left( {22 imes frac{{18}}{5}}
ight){ ext{km/sec}} cr
& { ext{ = 79}}{ ext{.2 km/hr}} cr} $$
& { ext{Speed}} cr
& { ext{ = }}left( {60 imes frac{5}{{18}}}
ight){ ext{m/sec}} cr
& { ext{ = }}frac{{50}}{3}{ ext{m/sec}} cr
& { ext{Time = 27 sec}}{ ext{.}} cr
& { ext{Let the length of the train be }}x{ ext{ metres}}{ ext{.}} cr
& { ext{Then,}}frac{{x + 200}}{{27}}{ ext{ = }}frac{{50}}{3}{ ext{ }} cr
& Leftrightarrow x + 200 = left( {frac{{50}}{3} imes 27}
ight) = 450 cr
& Leftrightarrow x = 450 - 200 = 250{ ext{ metres}} cr} $$
& { ext{Let the length of the train be x metres}}{ ext{.}} cr
& { ext{Then, length of the platform = (2}}x{ ext{) metres}}{ ext{.}} cr
& { ext{Speed of the train}} cr
& { ext{ = }}left( {90 imes frac{5}{{18}}}
ight)m/sec cr
& = 25m/sec cr
& herefore frac{{x + 2x}}{{25}} = 36 cr
& Rightarrow 3x = 900 cr
& Rightarrow x = 300 cr
& { ext{Hence, length of platform}} cr
& { ext{ = }}2x = left( {2 imes 300}
ight){ ext{m}} = 600{ ext{m}} cr} $$
& { ext{Speed = }}left( {frac{{150 + 300}}{{40.5}}}
ight)m/sec cr
& = left( {frac{{450}}{{40.5}} imes frac{{18}}{5}}
ight)km/hr cr
& = 40km/hr cr} $$
& { ext{Length of train}} cr
& { ext{ = 280 m }} cr
& { ext{Length of platform}} cr
& { ext{ = (3}} imes { ext{280) m = 840m}} cr
& herefore { ext{Speed of train}} cr
& { ext{ = }}left( {frac{{280 + 840}}{{50}}}
ight)m/sec cr
& = frac{{1120}}{{50}}m/sec cr
& = left( {frac{{1120}}{{50}} imes frac{{18}}{5}}
ight)km/hr cr
& = 80.64,km/hr cr} $$
& { ext{Speed of train relative to man}} cr
& { ext{ = }}left( {60 + 6}
ight){ ext{km/hr}} cr
& = 66,{ ext{km/hr}} cr
& = left( {66 imes frac{5}{{18}}}
ight)m/sec cr
& = left( {frac{{55}}{3}}
ight)m/sec cr
& herefore { ext{Time taken to pass the man}} cr
& = left( {110 imes frac{3}{{55}}}
ight)sec cr
& = 6,sec cr} $$
& { ext{Relative speed}} cr
& { ext{ = (72}} - { ext{60) km/hr}} cr
& { ext{ = 12 km/hr}} cr
& = left( {12 imes frac{5}{{18}}}
ight)m/sec cr
& = left( {frac{{10}}{3}}
ight)m/sec cr
& { ext{Total distance covered}} cr
& { ext{ = Sum of lengths of trains}} cr
& { ext{ = (240 + 240) m}} cr
& { ext{ = 480 m}} cr
& { ext{Time taken}} cr
& { ext{ = }}left( {480 imes frac{3}{{10}}}
ight)sec cr
& = 144sec cr
& = 2min ,24sec cr} $$
& { ext{Relative speed}} cr
& { ext{ = (60 + 90) km/hr}} cr
& { ext{ = }}left( {150 imes frac{5}{{18}}}
ight){ ext{m/sec}} cr
& { ext{ = }}left( {frac{{125}}{3}}
ight){ ext{m/sec}} cr
& { ext{Distance coverd}} cr
& { ext{ = (1}}{ ext{.10 + 0}}{ ext{.9)km}} cr
& { ext{ = 2 km}} cr
& { ext{ = 2000 m}}{ ext{}} cr
& { ext{Required time}} cr
& { ext{ = }}left( {2000 imes frac{3}{{125}}}
ight)sec cr
& = 48{ ext{ sec}}cr} $$
& { ext{Relative speed}} cr
& { ext{ = (80 + 55)km/hr}} cr
& { ext{ = 135 km/hr}} cr
& { ext{ = }}left( {135 imes frac{5}{{18}}}
ight)m/sec cr
& = left( {frac{{75}}{2}}
ight)m/sec cr
& { ext{Distance covered}} cr
& { ext{ = (120 + 90 + 90)m}} cr
& { ext{ = 300m}} cr
& { ext{Required time}} cr
& { ext{ = }}left( {300 imes frac{2}{{75}}}
ight)sec cr
& = 8sec cr} $$
& { ext{Relative speed}} cr
& { ext{ = (75 + 100)km/hr}} cr
& { ext{ = 175 km/hr}} cr
& { ext{Time taken to cover 175 km}} cr
& { ext{at relative speed = 1 hr}} cr
& herefore { ext{T = Time taken to cover 200 km}} cr
& { ext{ = }}left( {frac{1}{{175}} imes 200}
ight), ext{hr} cr
& = frac{8}{7}, ext{hr} cr
& = 1frac{1}{7}, ext{hr} cr} $$
& { ext{Speed of the train relative to man}} cr
& { ext{ = }}left( {frac{{240}}{{10}}}
ight){ ext{m/sec}} cr
& { ext{ = 24 m/sec}} cr
& { ext{ = }}left( {24 imes frac{{18}}{5}}
ight){ ext{ km/sec}} cr
& { ext{ = }}frac{{432}}{5}{ ext{km/hr}} cr
& { ext{Let the speed of the train be x kmph}}{ ext{.}} cr
& { ext{Then relative speed = }}left( {x + 3}
ight){ ext{kmph}} cr
& herefore x{ ext{ + 3 = }}frac{{432}}{5} cr
& Rightarrow x = frac{{432}}{5} - 3 cr
& Rightarrow x = frac{{417}}{5} cr
& ,,,,,,,,,,,,,, = 83.4,{ ext{kmph}} cr} $$
& { ext{Let the length of each train be }}x{ ext{ metres}} cr
& { ext{Then distance covered}} cr
& { ext{ = 2x metres}} cr
& { ext{Relative speed}} cr
& { ext{ = (46}} - { ext{36)km/hr}} cr
& { ext{ = }}left( {10 imes frac{5}{{18}}}
ight)m/sec cr
& = left( {frac{{25}}{9}}
ight)m/sec cr
& herefore frac{{2x}}{{36}} = frac{{25}}{9} Leftrightarrow 2x = 100 Leftrightarrow x = 50 cr} $$
& { ext{Speed of the train}} cr
& { ext{ = }}left( {frac{{120}}{{10}}}
ight){ ext{ m/sec}} cr
& { ext{ = 12 m/sec}} cr
& { ext{Speed of the second train}} cr
& { ext{ = }}left( {frac{{120}}{{15}}}
ight){ ext{ m/sec}} cr
& { ext{ = 8 m/sec}} cr
& { ext{Relative speed}} cr
& { ext{ = (12 + 8)m/sec}} cr
& { ext{ = 20 m/sec}} cr
& herefore { ext{Required time}} cr
& { ext{ = }}frac{{left( {120 + 120}
ight)}}{{20}},sec cr
& = 12,sec cr} $$
& { ext{Relative speed of the trains }} cr
& { ext{ = }}left( {frac{{100 + 200}}{{2 imes 60}}}
ight){ ext{m/sec}} cr
& { ext{ = }}left( {frac{5}{2}}
ight){ ext{m/sec}} cr
& { ext{Speed of train B}} cr
& { ext{ = 120 kmph}} cr
& = left( {120 imes frac{5}{{18}}}
ight){ ext{m/sec}} cr
& { ext{ = }}left( {frac{{100}}{3}}
ight){ ext{m/sec}} cr
& { ext{Let the speed of second train be }}x{ ext{ m/sec}} cr
& { ext{Then, }} frac{{100}}{3} - x = frac{5}{2} cr
& Rightarrow x = left( {frac{{100}}{3} - frac{5}{2}}
ight) cr
& ,,,,,,,,,,,,,, = left( {frac{{185}}{6}}
ight){ ext{m/sec}} cr
& herefore { ext{Speed of second train}} cr
& { ext{ = }}left( {frac{{185}}{6} imes frac{{18}}{5}}
ight){ ext{ kmph}} cr
& { ext{ = 111 kmph}} cr} $$
& { ext{Let the length of the train be }}x{ ext{ metres}} cr
& { ext{and its speed be }}y{ ext{ m/s}} cr
& { ext{Then,}} cr
& { ext{ }}frac{x}{{y - a}}{ ext{ = b}},,{ ext{and}}, cr
& ,frac{x}{{y - left( {a + 1}
ight)}} = left( {b + 1}
ight) cr
& Leftrightarrow { ext{ }}x{ ext{ = }}bleft( {y - a}
ight){ ext{ and}} cr
& ,,,,,,,,,,{ ext{ }}x = left( {b + 1}
ight)left( {y - a - 1}
ight) cr
& Leftrightarrow bleft( {y - a}
ight) = left( {b + 1}
ight)left( {y - a - 1}
ight) cr
& Leftrightarrow by - ba = by - ba - b + y - a - 1 cr
& Leftrightarrow y = left( {a + b + 1}
ight) cr} $$
& { ext{Distance travelled in 14 sec}} cr
& { ext{ = 50 + }}l cr
& { ext{Distance travelled in 10 sec}} cr
& { ext{ = }}l cr
& { ext{So speed of train}} cr
& { ext{ = }}frac{{50}}{{14 - 10}}{ ext{m/sec}} cr
& { ext{ = }}frac{{50}}{4} imes frac{{18}}{5}{ ext{km/hr}} cr
& { ext{ = 45 km/hr}} cr} $$
& { ext{Speed = 132 km/hr }} cr
& { ext{ = 132}} imes frac{5}{{18}}{ ext{m/sec}} cr
& { ext{ = }}frac{{110}}{3}m/sec cr
& T = frac{D}{S} cr
& ,,,,,, = frac{{110 + 165}}{{frac{{100}}{3}}} cr
& ,,,,,, = frac{{3left( {275}
ight)}}{{110}} cr
& ,,,,,, = 7.5sec cr} $$
& { ext{Speed of the train}} cr
& { ext{ = }}frac{{700 + 500}}{{10}} cr
& { ext{ = 120 ft/second}} cr} $$
& { ext{Ratio of speeds}} cr
& { ext{ = }}sqrt 4 :sqrt {6frac{1}{4}} cr
& = sqrt 4 :sqrt {frac{{25}}{4}} cr
& = 2:frac{5}{2} cr
& = 4:5 cr }$$ Let the speeds of the two trains be 4x and 5x km/hr respectively Then time taken by trains to meet each other $$eqalign{
& { ext{ = }}left( {frac{{270}}{{4x + 5x}}}
ight){ ext{hr}} cr
& { ext{ = }}left( {frac{{270}}{{9x}}}
ight){ ext{hr = }}left( {frac{{30}}{x}}
ight){ ext{hr}} cr
& { ext{Time taken by slower train to travel}} cr
& { ext{ 270 km = }}left( {frac{{270}}{{4x}}}
ight){ ext{hr}} cr
& herefore frac{{270}}{{4x}} = frac{{30}}{x} + 6frac{1}{4} cr
& Rightarrow frac{{270}}{{4x}} - frac{{30}}{x} = frac{{25}}{4} cr
& Rightarrow frac{{150}}{{4x}} = frac{{25}}{4} cr
& Rightarrow 100x = 600 cr
& Rightarrow x = 6 cr
& { ext{Hence speed of slower train}} cr
& { ext{ = 4}}x cr
& = ,24,{ ext{km/hr}} cr} $$