Question 1

Two stations P and Q are 110 km apart on a straight railway track. One train starts from station P at 7:00 a.m. and travels towards station Q at 20 km/h. Another train starts from station Q at 8:00 a.m. and travels towards station P at 25 km/h. At w…

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Introduction / Context: This question is a standard application of the relative speed concept, where two trains start at different times from two stations and travel towards each other. It checks your understanding of how to handle different starting times and how to set up a simple linear equation based on distance, speed, and time. Given Data / Assumptions: Distance between stations P and Q = 110 km. Train from P leaves at 7:00 a.m. with speed 20 km/h. Train from Q leaves at 8:00 a.m. with speed 25 km/h. Both trains travel along the same straight track towards each other. We must find the clock time when they meet. Concept / Approach: When two objects move towards each other, their relative speed is the sum of their individual speeds, but here they start at different times. The standard method is to measure time from one train's start and adjust for the delayed start of the second train. Let the time in hours from 7:00 a.m. to the meeting instant be t. Then the first train travels for t hours and the second train travels for t minus 1 hours, because it starts one hour later. The sum of their distances will be equal to 110 km. Step-by-Step Solution: Step 1: Let t be the time in hours after 7:00 a.m. when the trains meet.Step 2: Distance covered by the first train from P = 20 * t.Step 3: The second train from Q starts at 8:00 a.m., so it travels for (t - 1) hours. Its distance = 25 * (t - 1).Step 4: At the meeting point, total distance covered = 110 km, so 20 * t + 25 * (t - 1) = 110.Step 5: Simplify: 20t + 25t - 25 = 110, giving 45t - 25 = 110, hence 45t = 135 and t = 3 hours.Step 6: Time of meeting = 7:00 a.m. + 3 hours = 10:00 a.m. Verification / Alternative check: In three hours, the first train covers 20 * 3 = 60 km.The second train travels for 2 hours (from 8:00 a.m. to 10:00 a.m.) and covers 25 * 2 = 50 km.Total distance = 60 + 50 = 110 km, which matches the distance between the stations. Why Other Options Are Wrong: Times like 9:30 a.m. or 9:15 a.m. arise if you incorrectly assume both trains start together or mishandle the one-hour delay. The option 8:45 a.m. is too early, leaving insufficient time for either train to cover the required distances. The option 10:30 a.m. would cause the trains to cover more than 110 km combined, implying they would have already met earlier. Common Pitfalls: Many candidates forget to account for the late start of the second train and treat the problem as if both trains started at 7:00 a.m. Another frequent error is to use relative speed directly without adjusting time, which works only when starting times are the same. Correctly defining the variable t and writing the distance equation is the safest way to avoid confusion. Final Answer: The two trains will meet at 10:00 a.m.

Question 2

Two trains of equal length are running in opposite directions on parallel tracks. They pass a fixed pole in 18 seconds and 12 seconds respectively. In how many seconds will the two trains completely cross each other when running in opposite directio…

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Introduction / Context: This is another standard trains question dealing with two trains of equal length moving in opposite directions. It tests understanding of how to use times taken to pass a fixed point to deduce speeds, and then how to use relative speed to determine the time required for the trains to completely cross each other. Given Data / Assumptions: Both trains have the same length, say L metres. The first train passes a pole in 18 seconds. The second train passes a pole in 12 seconds. The trains run in opposite directions on parallel tracks with uniform speeds. We must find the time taken for them to cross each other completely. Concept / Approach: When a train passes a pole, the distance covered is its own length. Therefore, for each train, speed equals length divided by time. When two trains cross each other while moving in opposite directions, the effective distance to be covered is the sum of their lengths, and the relative speed is the sum of their individual speeds. We express both speeds in terms of the common length L to find the required crossing time. Step-by-Step Solution: Step 1: Let the length of each train be L metres.Step 2: Speed of first train = L / 18 metres per second.Step 3: Speed of second train = L / 12 metres per second.Step 4: Relative speed when moving in opposite directions = L / 18 + L / 12.Step 5: Simplify the relative speed: L / 18 + L / 12 = (2L + 3L) / 36 = 5L / 36 metres per second.Step 6: Effective distance to be covered while crossing each other = L + L = 2L metres.Step 7: Time to cross each other = distance / relative speed = 2L / (5L / 36) = 2L * 36 / (5L) = 72 / 5 seconds = 14.4 seconds. Verification / Alternative check: Pick a convenient length, for example L = 180 metres.Then speed of first train = 180 / 18 = 10 metres per second, and speed of second train = 180 / 12 = 15 metres per second.Relative speed = 10 + 15 = 25 metres per second and total distance = 360 metres.Time = 360 / 25 = 14.4 seconds, confirming the result. Why Other Options Are Wrong: Times such as 15.5 seconds, 18.8 seconds or 20.2 seconds arise from algebraic errors or from using the difference of speeds instead of their sum. The value 16 seconds is an approximate guess and does not satisfy the correct distance and speed calculations. Common Pitfalls: A common mistake is to forget that the total distance during crossing is the sum of the lengths of both trains. Another error is confusing relative speed in the case of objects moving in the same direction (difference of speeds) with those moving in opposite directions (sum of speeds). Also, candidates sometimes cancel the length L incorrectly in intermediate steps, leading to wrong numerical values. Final Answer: The two trains will completely cross each other in 14.4 seconds.

Question 3

A train is travelling at a constant speed of 75 miles per hour and enters a tunnel that is 3.5 miles long. The train itself is 0.25 mile long. How long, in minutes, does it take from the moment the front of the train enters the tunnel until the mome…

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Introduction / Context: This tunnel problem tests understanding of how much distance a train must cover to clear a tunnel completely. It also checks correct handling of miles and hours units and converting the final answer into minutes. Such questions are common in exams because they combine geometry (lengths) with uniform motion concepts. Given Data / Assumptions: Speed of the train = 75 miles per hour. Length of the tunnel = 3.5 miles. Length of the train = 0.25 mile. The train moves at a constant speed with no acceleration or deceleration inside the tunnel. We need the total time from front entering until rear leaving, expressed in minutes. Concept / Approach: To clear the tunnel completely, the rear end of the train must move past the far end of the tunnel. Therefore, the total distance traveled by the train's front from entry to final exit is the sum of the tunnel length and the train length. Once we know this effective distance and the constant speed, we can find the time using the basic formula time = distance / speed. Finally, we convert the time from hours to minutes by multiplying by 60. Step-by-Step Solution: Step 1: Effective distance to be covered = tunnel length + train length.Step 2: Tunnel length = 3.5 miles, train length = 0.25 mile, so total distance = 3.5 + 0.25 = 3.75 miles.Step 3: Speed of the train = 75 miles per hour.Step 4: Time in hours = distance / speed = 3.75 / 75.Step 5: Simplify 3.75 / 75 = 0.05 hours.Step 6: Convert hours to minutes: time in minutes = 0.05 * 60 = 3 minutes. Verification / Alternative check: We can think of 75 miles per hour as 1.25 miles per minute, since 75 / 60 = 1.25.If the train needs to cover 3.75 miles, time in minutes = 3.75 / 1.25 = 3 minutes.This alternative view confirms that the previous calculation is correct. Why Other Options Are Wrong: Values like 2.5 minutes or 3.2 minutes come from rounding too early or using incorrect effective distance. The option 3.5 minutes could result from erroneously adding extra distance or misinterpreting the required time. The value 4 minutes is clearly larger and would imply a lower speed than given in the question. Only 3 minutes satisfies the precise distance and speed relationship. Common Pitfalls: Candidates sometimes forget to add the train length to the tunnel length and mistakenly use only the tunnel's length. Others confuse miles per hour with miles per minute and do not convert correctly to minutes. It is also easy to mix up when to multiply or divide by 60 during unit conversion. Keeping track of units at every step helps prevent these errors. Final Answer: The train takes 3 minutes to completely pass through the tunnel.

Question 4

A train speeds past a fixed pole in 15 seconds and passes completely over a platform that is 100 metres long in 25 seconds. What is the length of the train in metres?

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Introduction / Context: This is a classic train and platform question. It checks whether you can distinguish between the distance covered when a train passes a fixed point (such as a pole) and when it passes an extended object like a platform. It also tests your skill in forming equations from the standard speed, distance and time relationship. Given Data / Assumptions: The train passes a pole in 15 seconds. The train passes a platform 100 metres long in 25 seconds. The train moves at constant speed along a straight track. We must find the length of the train in metres. Concept / Approach: When a train passes a pole, the distance travelled during that time is equal to the train's own length. When the same train passes a platform, the distance travelled is equal to the sum of the train length and the platform length. Let the length of the train be L metres and its speed be v metres per second. From the pole information we can express v in terms of L, substitute into the platform equation, and then solve for L. Step-by-Step Solution: Step 1: Let the train length be L metres and its speed be v metres per second.Step 2: From passing the pole, L / v = 15, so v = L / 15.Step 3: For the platform, total distance to be covered = L + 100 metres, and time taken = 25 seconds.Step 4: So (L + 100) / v = 25. Substitute v = L / 15.Step 5: This gives (L + 100) / (L / 15) = 25, which simplifies to 15 * (L + 100) / L = 25.Step 6: Multiply out: 15L + 1500 = 25L, so 1500 = 10L, hence L = 150 metres. Verification / Alternative check: If L = 150 metres, then speed v = L / 15 = 150 / 15 = 10 metres per second.Time to pass platform = (L + 100) / v = (150 + 100) / 10 = 250 / 10 = 25 seconds, which matches the given data.Time to pass pole = L / v = 150 / 10 = 15 seconds, also matching the information given. Why Other Options Are Wrong: 50 metres or 120 metres are far too small; they would not satisfy both time conditions simultaneously. The value 200 metres is too large and leads to inconsistent times when checked back. The option marked as data inadequate is incorrect because there is enough information to solve uniquely for the train length using two equations in two variables. Common Pitfalls: Students sometimes forget to add the platform length to the train length while computing the distance during platform crossing. Another common error is to confuse the roles of time and speed in the equations or to cancel terms incorrectly. Writing each equation carefully and substituting step by step is the safest method. Final Answer: The length of the train is 150 metres.

Question 5

A train is moving along a straight track and crosses a man who is running on a platform at a speed of 8 km/h in the same direction as the train. The train takes 9 seconds to completely pass the man. If the speed of the train is 74 km/h, what is the …

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Introduction / Context: This problem deals with relative speed when a fast object such as a train overtakes a slower moving person on a platform. It examines your ability to find the relative speed when both move in the same direction and then use that relative speed to compute the length of the train from the time taken to overtake. Given Data / Assumptions: Speed of the train = 74 km/h. Speed of the man running on the platform = 8 km/h. The man and the train move in the same direction. The train takes 9 seconds to completely pass the man. We need to determine the length of the train in metres. Concept / Approach: When two objects move in the same direction, the relative speed is the difference between their speeds. The length of the train is the distance covered by the train relative to the man during the time it takes to overtake him. We first compute the relative speed in km/h, convert it into metres per second, and then multiply by the time in seconds to obtain the length of the train in metres. Step-by-Step Solution: Step 1: Relative speed when moving in the same direction = speed of train - speed of man = 74 - 8 = 66 km/h.Step 2: Convert 66 km/h into metres per second. Use 1 km/h = 5 / 18 metres per second.Step 3: Relative speed in metres per second = 66 * 5 / 18.Step 4: Simplify: 66 / 18 = 11 / 3, so relative speed = (11 / 3) * 5 = 55 / 3 metres per second.Step 5: Time taken to overtake = 9 seconds, so distance covered relative to the man (which equals the train length) = relative speed * time = (55 / 3) * 9 = 55 * 3 = 165 metres. Verification / Alternative check: If the train length is 165 metres and relative speed is 55 / 3 metres per second, then time = distance / speed = 165 / (55 / 3) = 165 * 3 / 55 = 9 seconds.This matches the time given in the question, confirming that the computed length is correct. Why Other Options Are Wrong: Values such as 156 metres, 188 metres or 202 metres arise if you approximate relative speed incorrectly or mishandle unit conversions. The value 140 metres is too small and would imply a smaller overtaking time than 9 seconds given the same speeds. Only 165 metres is consistent with the data and formulae. Common Pitfalls: A common mistake is to add the speeds instead of subtracting them when both objects move in the same direction. Learners may also forget to convert km/h to metres per second before multiplying by time in seconds, which leads to a result in mixed units. Keeping units consistent and remembering that relative speed is the difference in the same direction prevent these errors. Final Answer: The length of the train is 165 metres.

Question 6

Two trains of equal length, each 120 metres long, take 10 seconds and 15 seconds respectively to cross a fixed post. If they are now travelling on parallel tracks in opposite directions, in how many seconds will they completely cross each other?

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Introduction / Context: This problem is another example of trains moving on parallel tracks and crossing each other. It uses the information about the times taken to pass a fixed post to deduce the speeds of the trains and then applies relative speed to find how long they take to cross each other fully when travelling in opposite directions. Given Data / Assumptions: Each train has length 120 metres. The first train passes a post in 10 seconds. The second train passes a post in 15 seconds. Both trains move with uniform speeds on parallel tracks. The trains then run in opposite directions and we must find the time to cross completely. Concept / Approach: When a train passes a post, the distance travelled is equal to the train's own length. So from each given time we can find the speed of that train in metres per second. When the trains run in opposite directions, the relative speed is the sum of their speeds and the effective distance to be covered while crossing is the total of both lengths. Time is then found as distance divided by relative speed. Step-by-Step Solution: Step 1: Speed of first train = distance / time = 120 / 10 = 12 metres per second.Step 2: Speed of second train = 120 / 15 = 8 metres per second.Step 3: Relative speed when moving in opposite directions = 12 + 8 = 20 metres per second.Step 4: Total distance to be covered while crossing = 120 + 120 = 240 metres.Step 5: Time to cross each other completely = distance / speed = 240 / 20 = 12 seconds. Verification / Alternative check: If the trains cross in 12 seconds at relative speed 20 metres per second, the total distance covered is 20 * 12 = 240 metres.This matches the combined length of the trains (120 + 120), so the time is consistent with the given lengths and speeds. Why Other Options Are Wrong: The options 10 seconds or 15 seconds are simply the times taken to pass a post, not the time to cross another moving train. The option 20 seconds assumes a smaller relative speed, which is incorrect in opposite directions. The option 25 seconds would be even slower and does not match the calculated relative speed. Common Pitfalls: Some learners forget to add the lengths of both trains and instead use only one length, which leads to half the actual time. Others mistakenly use the difference of speeds rather than their sum, which is only appropriate when both trains move in the same direction. Clear thinking about relative motion directions is essential to avoid these missteps. Final Answer: The two trains will completely cross each other in 12 seconds.

Question 7

A train that is 270 metres long is running at a speed of 120 km/h. It crosses another train running in the opposite direction at a speed of 80 km/h in 9 seconds. What is the length of the other train in metres?

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Introduction / Context: This question focuses on relative speed and the total effective distance when two trains running in opposite directions cross each other. It tests whether you can combine speeds given in km/h, convert them to metres per second, and use the crossing time to find the unknown length of the second train. Given Data / Assumptions: Length of first train = 270 metres. Speed of first train = 120 km/h. Speed of second train = 80 km/h. The trains move in opposite directions on parallel tracks. Time taken to cross each other completely = 9 seconds. We need the length of the second train. Concept / Approach: When two trains run in opposite directions, their relative speed is the sum of their individual speeds. The distance that must be covered for them to cross each other completely is the sum of their lengths. We first convert both speeds from km/h to metres per second, add them to obtain the relative speed, multiply by the given time to get total distance, and finally subtract the known train length from this total to find the unknown length. Step-by-Step Solution: Step 1: Convert speeds to metres per second using 1 km/h = 5 / 18 metres per second.Step 2: Speed of first train = 120 * 5 / 18 = 600 / 18 = 100 / 3 metres per second.Step 3: Speed of second train = 80 * 5 / 18 = 400 / 18 = 200 / 9 metres per second.Step 4: Relative speed = 100 / 3 + 200 / 9 = (300 + 200) / 9 = 500 / 9 metres per second.Step 5: Total distance covered while crossing = relative speed * time = (500 / 9) * 9 = 500 metres.Step 6: Let the length of the second train be L metres. Then 270 + L = 500, so L = 500 - 270 = 230 metres. Verification / Alternative check: If the second train is 230 metres long, total length is 270 + 230 = 500 metres.With relative speed 500 / 9 metres per second, time = distance / speed = 500 / (500 / 9) = 9 seconds.This equals the given crossing time, confirming the correctness of the length. Why Other Options Are Wrong: Values such as 240 metres, 260 metres or 320 metres do not satisfy the distance = speed * time relationship when checked back with the same data. For example, a longer train such as 320 metres would yield a total length larger than 500 metres and hence a longer crossing time than 9 seconds. The value 200 metres is too small and leads to a shorter crossing time than stated. Common Pitfalls: Common mistakes include forgetting to convert speeds from km/h to metres per second, adding lengths incorrectly, or failing to use the sum of speeds in opposite directions. Some candidates also mix up the roles of the two trains and subtract lengths instead of adding them. Keeping the relationships clear and checking units at each stage helps avoid such errors. Final Answer: The length of the other train is 230 metres.

Question 8

A passenger train runs between stations K and L. It covers the distance 40 minutes faster than a goods train that runs between the same stations. If the average speed of the passenger train is 50 km/h and that of the goods train is 30 km/h, what is …

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Introduction / Context: This question compares two trains travelling the same route at different speeds and asks for the distance between the stations based on the difference in travel times. It is a straightforward application of the distance = speed * time relationship combined with the idea of a fixed time difference between two journeys. Given Data / Assumptions: Passenger train speed = 50 km/h. Goods train speed = 30 km/h. Passenger train takes 40 minutes less than the goods train to cover the same distance. Forty minutes is equal to 2 / 3 hour. The distance between stations K and L is the same for both trains. Concept / Approach: Let the distance between stations K and L be D kilometres. Then the time taken by the passenger train is D / 50 hours, and the time taken by the goods train is D / 30 hours. The difference between these times is given as 2 / 3 hour. We write an equation representing this time difference and solve it for D, obtaining the required distance. Step-by-Step Solution: Step 1: Let D be the distance in kilometres between K and L.Step 2: Time taken by passenger train = D / 50 hours.Step 3: Time taken by goods train = D / 30 hours.Step 4: Time difference = (D / 30) - (D / 50) = 2 / 3 hour, because the goods train is slower and takes longer.Step 5: Compute the difference: D / 30 - D / 50 = D * (1 / 30 - 1 / 50) = D * ((5 - 3) / 150) = D * (2 / 150) = D / 75.Step 6: So D / 75 = 2 / 3, which gives D = 75 * (2 / 3) = 50 kilometres. Verification / Alternative check: For D = 50 km, time of passenger train = 50 / 50 = 1 hour.Time of goods train = 50 / 30 hours = 5 / 3 hours ≈ 1.67 hours.Difference in time = 5 / 3 - 1 = 2 / 3 hour, which is exactly 40 minutes, matching the problem statement. Why Other Options Are Wrong: Distances such as 44 km, 46 km, 48 km or 52 km do not produce a time difference of 40 minutes when combined with speeds of 50 km/h and 30 km/h. If you check them using the same method, you will find that the time differences are either smaller or larger than 2 / 3 hour, so they cannot be correct. Common Pitfalls: Some candidates incorrectly treat 40 minutes as 0.4 hour instead of 2 / 3 hour. Others accidentally reverse the time difference and write D / 50 - D / 30 instead of D / 30 - D / 50. Another error is to convert units inconsistently or to forget that both trains travel the same distance. Writing down the time expressions clearly and handling the fraction arithmetic carefully avoids these issues. Final Answer: The distance between stations K and L is 50 km.

Question 9

A train travelling at 60 km/h catches up with a slower train travelling in the same direction. From the moment it draws level with the slower train, it leaves the slower train 120 metres behind in 18 seconds. What is the speed of the slower train in…

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Introduction / Context: This question deals with a faster train overtaking a slower train moving in the same direction. It tests understanding of relative speed in the same direction and how to convert a relative distance gained over a known time into the difference between speeds. From this, the speed of the slower train can be determined. Given Data / Assumptions: Speed of faster train = 60 km/h. Trains move in the same direction on parallel tracks. In 18 seconds, after drawing level, the faster train is 120 metres ahead of the slower train. The distance gained relative to the slower train in that time is 120 metres. We must find the speed of the slower train in km/h. Concept / Approach: When two objects move in the same direction, the relative speed is the difference between their speeds. The relative distance covered in a certain time is equal to this relative speed multiplied by the time. Here, the faster train gains 120 metres on the slower one in 18 seconds, so the relative speed in metres per second is 120 / 18. We then convert that relative speed into km/h and subtract it from the speed of the faster train to obtain the speed of the slower one. Step-by-Step Solution: Step 1: Relative distance gained = 120 metres in 18 seconds.Step 2: Relative speed in metres per second = 120 / 18 = 20 / 3 metres per second.Step 3: Convert relative speed to km/h using 1 metre per second = 18 / 5 km/h.Step 4: Relative speed in km/h = (20 / 3) * 18 / 5 = (20 * 18) / (3 * 5) = 360 / 15 = 24 km/h.Step 5: Let speed of slower train be v km/h. Then 60 - v = 24, so v = 60 - 24 = 36 km/h. Verification / Alternative check: If the slower train runs at 36 km/h, then the speed difference = 60 - 36 = 24 km/h.Convert 24 km/h to metres per second: 24 * 5 / 18 = 20 / 3 metres per second.In 18 seconds, distance gained = (20 / 3) * 18 = 120 metres, which matches the question statement. Why Other Options Are Wrong: Values like 42 km/h or 44 km/h result from arithmetic mistakes when converting units or manipulating equations. The option 72 km/h is larger than the speed of the faster train and is impossible. A slower train speed of 48 km/h gives a relative speed of only 12 km/h, which does not produce a gain of 120 metres in 18 seconds. Common Pitfalls: Some learners treat the distance 120 metres as the length of a train rather than the relative distance gained, leading to a different setup. Others forget to convert between metres per second and km/h or use the wrong factor 5 / 18 instead of 18 / 5 at the wrong step. Carefully distinguishing between absolute and relative motion and checking unit consistency prevent such errors. Final Answer: The speed of the slower train is 36 km/h.

Question 10

Two trains, 140 metres and 160 metres long respectively, run on parallel tracks in opposite directions. Their speeds are 60 km/h and 40 km/h respectively. How many seconds will they take to completely cross each other?

Accepted Answer

Introduction / Context: This problem again checks understanding of relative speed when two trains move in opposite directions. The twist is that the two trains have different lengths and different speeds. The key skills tested are unit conversion and correctly determining the total distance and relative speed. Given Data / Assumptions: Length of first train = 140 metres. Length of second train = 160 metres. Speed of first train = 60 km/h. Speed of second train = 40 km/h. The trains move in opposite directions on parallel tracks. We must find the time taken for them to completely cross each other. Concept / Approach: When trains move in opposite directions, the total distance to be covered to cross completely is the sum of their lengths. Their relative speed is the sum of their individual speeds. We convert the speeds from km/h to metres per second, add them to get the relative speed, then divide the total distance by this relative speed to obtain the required crossing time in seconds. Step-by-Step Solution: Step 1: Total distance to be covered while crossing = 140 + 160 = 300 metres.Step 2: Convert speeds to metres per second: 60 km/h = 60 * 5 / 18 = 50 / 3 metres per second.Step 3: Similarly, 40 km/h = 40 * 5 / 18 = 100 / 9 metres per second.Step 4: Relative speed in opposite directions = 50 / 3 + 100 / 9 = (150 + 100) / 9 = 250 / 9 metres per second.Step 5: Time = distance / speed = 300 / (250 / 9) = 300 * 9 / 250 = 2700 / 250 = 10.8 seconds. Verification / Alternative check: We can approximate: 60 km/h and 40 km/h sum to 100 km/h, roughly equal to 27.78 metres per second.Then time ≈ 300 / 27.78 ≈ 10.8 seconds, consistent with the exact calculation. Why Other Options Are Wrong: Values like 9 seconds or 9.6 seconds are too short and correspond to a higher relative speed than possible with 60 km/h and 40 km/h. The options 10 seconds or 12 seconds are rough guesses that do not match the exact ratio of distance to relative speed when recalculated precisely. Common Pitfalls: One common error is to forget to add the lengths of both trains and use only a single length as the total distance. Another is to use the difference of speeds instead of the sum when trains move in opposite directions. Also, some students forget to convert from km/h to metres per second, which leads to inconsistent units and wrong times. Final Answer: The trains will completely cross each other in 10.8 seconds.

Question 11

A train is travelling at 48 km/h and crosses another train, having half of its own length and travelling in the opposite direction at 42 km/h, in 12 seconds. The same train then passes completely over a bridge in 45 seconds. What is the length of th…

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Introduction / Context: This composite train problem first uses relative speed in opposite directions to determine the length of a train, and then uses that length with the train's speed and the time to cross a bridge to find the bridge length. It tests multi step reasoning, unit conversion, and careful algebra. Given Data / Assumptions: Speed of first train = 48 km/h. Speed of second train = 42 km/h. Length of second train = half the length of the first train. The trains cross each other in 12 seconds while travelling in opposite directions. The first train passes a bridge in 45 seconds at 48 km/h. We must find the length of the bridge in metres. Concept / Approach: Let the length of the first train be L metres, so the second train has length L / 2 metres. In opposite directions, relative speed is the sum of their speeds, and the total distance during crossing is L + L / 2. We first determine L from this information. Then, treating the bridge as a fixed object, the distance covered while passing the bridge is L plus the bridge length. Using the constant speed of 48 km/h and the 45 second crossing time, we solve for the bridge length. Step-by-Step Solution: Step 1: Convert speeds to metres per second: 48 km/h = 48 * 5 / 18 = 40 / 3 metres per second, and 42 km/h = 42 * 5 / 18 = 35 / 3 metres per second.Step 2: Relative speed in opposite directions = 40 / 3 + 35 / 3 = 75 / 3 = 25 metres per second.Step 3: Total distance when trains cross each other = L + L / 2 = 3L / 2 metres.Step 4: Time for crossing = 12 seconds, so 3L / 2 = 25 * 12 = 300, which gives L = 200 metres.Step 5: Now consider the first train passing a bridge. Distance covered = L + bridge length = 200 + B, speed = 48 km/h = 40 / 3 metres per second, time = 45 seconds.Step 6: Use distance = speed * time: 200 + B = (40 / 3) * 45 = 40 * 15 = 600, giving B = 600 - 200 = 400 metres. Verification / Alternative check: Check crossing of trains: with L = 200, second train length = 100 metres. Total distance = 300 metres, relative speed = 25 metres per second, time = 300 / 25 = 12 seconds as required.Check bridge crossing: with bridge length 400 metres, total distance = 600 metres and speed = 40 / 3 metres per second, time = 600 / (40 / 3) = 600 * 3 / 40 = 45 seconds. Why Other Options Are Wrong: Bridge lengths such as 250 metres, 320 metres, 390 metres or 280 metres do not satisfy both the crossing and bridge conditions when verified. Some values correspond to using wrong relative speeds or incorrect time conversions. Only 400 metres is consistent with all parts of the question. Common Pitfalls: Learners may forget that the second train is only half as long as the first, leading to an incorrect total distance during the crossing. Another common error is to use the difference of speeds instead of the sum for opposite directions. Mistakes can also occur when converting km/h to metres per second or when solving the equation 3L / 2 = 300. Organising the problem in two clear stages helps to avoid confusion. Final Answer: The length of the bridge is 400 metres.

Question 12

A train covers the distance between station A and station B in 45 minutes at its usual speed. If the speed of the train is reduced by 5 km/h, it takes 48 minutes to cover the same distance. What is the distance between stations A and B in kilometres?

Accepted Answer

Introduction / Context: This problem explores how changes in speed affect travel time over a fixed distance. It tests the ability to convert minutes into hours, to set up equations based on distance = speed * time, and to solve for the unknown distance when two different speed and time combinations are given. Given Data / Assumptions: At original speed, the train covers the distance in 45 minutes. At reduced speed, which is 5 km/h less, it covers the same distance in 48 minutes. We must find the distance between stations A and B. Speeds are constant in each scenario and the track is straight. Concept / Approach: Let the original speed be v km/h and the distance be D kilometres. Time in hours is obtained by dividing minutes by 60. The first equation is D = v * (45 / 60), and the second equation is D = (v - 5) * (48 / 60). Since both expressions equal D, we can equate them and solve for v. After finding v, we substitute back into either expression to determine D. Step-by-Step Solution: Step 1: Convert times into hours: 45 minutes = 45 / 60 = 3 / 4 hour, and 48 minutes = 48 / 60 = 4 / 5 hour.Step 2: Let original speed be v km/h and distance be D km.Step 3: From the first condition, D = v * 3 / 4.Step 4: From the second condition, D = (v - 5) * 4 / 5.Step 5: Equate the two expressions for D: v * 3 / 4 = (v - 5) * 4 / 5.Step 6: Cross multiply: 5 * 3v = 4 * 4 * (v - 5), which gives 15v = 16v - 80, so v = 80 km/h.Step 7: Substitute back to find D: D = v * 3 / 4 = 80 * 3 / 4 = 60 km. Verification / Alternative check: At 80 km/h, time to travel 60 km = 60 / 80 hours = 3 / 4 hour = 45 minutes, matching the first condition.At reduced speed 75 km/h, time to travel 60 km = 60 / 75 hours = 4 / 5 hour = 48 minutes, matching the second condition. Why Other Options Are Wrong: Distances like 32 km, 45 km, 50 km or 80 km do not satisfy both travel times when checked with the given speed changes. If you plug them into the distance = speed * time relationship, at least one of the 45 minute or 48 minute conditions will fail. Common Pitfalls: Some learners forget to convert minutes to hours, which leads to incorrect distances. Others incorrectly add 5 km/h instead of subtracting it or mis-handle the cross multiplication step. Writing down both equations clearly and working through the algebra carefully is essential for such questions. Final Answer: The distance between stations A and B is 60 km.

Question 13

A goods train runs at a speed of 72 km/h and crosses a platform that is 250 metres long in 26 seconds. What is the length of the goods train in metres?

Accepted Answer

Introduction / Context: This is a straightforward train and platform question where the length of the train has to be found from its speed, the time taken to pass the platform, and the known platform length. It reinforces the relationship between speed, distance and time and proper unit conversion from km/h to metres per second. Given Data / Assumptions: Speed of the goods train = 72 km/h. Length of the platform = 250 metres. Time taken to cross the platform completely = 26 seconds. The train moves at uniform speed. We must determine the length of the train. Concept / Approach: When a train passes a platform, the distance it travels from the instant its front end reaches the platform to the instant its rear end leaves the platform is equal to the sum of the train length and the platform length. We first convert the train's speed into metres per second, then multiply by the time to get the total distance travelled during crossing, and finally subtract the platform length to obtain the train length. Step-by-Step Solution: Step 1: Convert speed from km/h to metres per second: 72 km/h = 72 * 5 / 18 = 20 metres per second.Step 2: Total distance covered while crossing the platform = speed * time = 20 * 26 = 520 metres.Step 3: Let the train length be L metres. Then L + 250 = 520.Step 4: Solve for L: L = 520 - 250 = 270 metres. Verification / Alternative check: Using L = 270 metres, total distance while crossing platform = 270 + 250 = 520 metres.At 20 metres per second, time = 520 / 20 = 26 seconds, which exactly matches the given time. Why Other Options Are Wrong: Train lengths such as 230 metres, 240 metres, 260 metres or 280 metres do not result in a total distance of 520 metres when added to the platform length, and therefore do not produce the correct crossing time of 26 seconds at 72 km/h. When checked, each one either gives a time shorter or longer than 26 seconds. Common Pitfalls: Some students forget to convert speed from km/h into metres per second and directly multiply by time in seconds, which mixes inconsistent units. Others may mistakenly subtract the platform length first or confuse the formula, using distance minus platform instead of distance equals train length plus platform length. Keeping the interpretation of distance clear eliminates these issues. Final Answer: The length of the goods train is 270 metres.

Question 14

The difference between the times taken by two trains to travel a distance of 350 km is 2 hours 20 minutes. The difference between their speeds is 5 km/h. What is the speed of the faster train in km/h?

Accepted Answer

Introduction / Context: This problem involves two trains travelling the same fixed distance at different speeds. The difference in their travel times and speeds is known. The question tests your ability to translate this information into a simple algebraic equation using the distance = speed * time relationship and to solve it to find the faster speed. Given Data / Assumptions: Common distance travelled by both trains = 350 km. Difference in their speeds = 5 km/h. Difference in their times = 2 hours 20 minutes. Two hours 20 minutes equals 7 / 3 hours. We must find the speed of the faster train. Concept / Approach: Let the speed of the slower train be s km/h. Then the speed of the faster train is s + 5 km/h. Time taken by the slower train is 350 / s hours and by the faster train is 350 / (s + 5) hours. The difference in times equals 7 / 3 hours. Setting up the equation 350 / s - 350 / (s + 5) = 7 / 3 and solving for s gives the slower speed. Adding 5 then yields the faster speed. Step-by-Step Solution: Step 1: Let speed of slower train be s km/h, faster train speed = s + 5 km/h.Step 2: Time taken by slower train = 350 / s hours.Step 3: Time taken by faster train = 350 / (s + 5) hours.Step 4: Given that 350 / s - 350 / (s + 5) = 7 / 3.Step 5: Factor out 350: 350 * (1 / s - 1 / (s + 5)) = 7 / 3.Step 6: Compute the bracket: 1 / s - 1 / (s + 5) = (s + 5 - s) / (s * (s + 5)) = 5 / (s * (s + 5)).Step 7: So 350 * 5 / (s * (s + 5)) = 7 / 3, which simplifies to 1750 / (s * (s + 5)) = 7 / 3.Step 8: Cross multiply to get 1750 * 3 = 7 * s * (s + 5) which simplifies to 5250 = 7s^2 + 35s.Step 9: Divide by 7: 750 = s^2 + 5s. So s^2 + 5s - 750 = 0.Step 10: Solve the quadratic: s = 25 or a negative value. We take s = 25 km/h as valid.Step 11: Faster train speed = s + 5 = 25 + 5 = 30 km/h. Verification / Alternative check: Time for slower train at 25 km/h = 350 / 25 = 14 hours.Time for faster train at 30 km/h = 350 / 30 ≈ 11.67 hours = 35 / 3 hours.Difference = 14 - 35 / 3 = 42 / 3 - 35 / 3 = 7 / 3 hours = 2 hours 20 minutes, which matches the question. Why Other Options Are Wrong: Speeds such as 36 km/h, 34 km/h or 40 km/h do not yield a time difference of 2 hours 20 minutes when used with the same 5 km/h gap. The option 28 km/h is less than the calculated faster speed and would correspond to a different pair of speeds. Only 30 km/h satisfies both the distance and time conditions. Common Pitfalls: Learners sometimes misconvert 2 hours 20 minutes into 2.2 hours instead of 7 / 3 hours. Others incorrectly set up the equation, reversing the time difference or forgetting that the slower train takes longer. Algebraic mistakes in solving the quadratic equation can also lead to incorrect speeds. Careful handling of fractions and a quick check by back substitution help to avoid these errors. Final Answer: The speed of the faster train is 30 km/h.

Question 15

A train 800 metres long is running at a speed of 78 km/h. It passes completely through a tunnel in 1 minute. What is the length of the tunnel in metres?

Accepted Answer

Introduction / Context: This tunnel problem is a direct application of uniform motion. It tests whether you can compute the total distance a train travels while it completely passes through a tunnel and then subtract the train length to obtain the tunnel length. It also requires correct conversion from km/h to metres per second and accurate use of the time in seconds. Given Data / Assumptions: Length of the train = 800 metres. Speed of the train = 78 km/h. Time taken to completely pass through the tunnel = 1 minute = 60 seconds. The train travels at a constant speed. We must find the length of the tunnel. Concept / Approach: To clear the tunnel completely, the train must travel a distance equal to the sum of its own length and the tunnel length. We first convert the speed from km/h to metres per second. Then, using distance = speed * time, we find the total distance covered while inside the tunnel. Finally, subtracting the train length from this total distance yields the tunnel length. Step-by-Step Solution: Step 1: Convert speed from km/h to metres per second: 78 km/h = 78 * 5 / 18 metres per second.Step 2: Compute 78 * 5 / 18 = 390 / 18 = 65 / 3 metres per second.Step 3: Time taken to pass the tunnel completely = 1 minute = 60 seconds.Step 4: Total distance travelled while passing the tunnel = speed * time = (65 / 3) * 60 = 65 * 20 = 1300 metres.Step 5: Let tunnel length be T metres. Then total distance = train length + tunnel length = 800 + T.Step 6: So 800 + T = 1300, giving T = 1300 - 800 = 500 metres. Verification / Alternative check: If the tunnel length is 500 metres, total distance to clear it is 800 + 500 = 1300 metres.At 65 / 3 metres per second, time = 1300 / (65 / 3) = 1300 * 3 / 65 = 60 seconds, which is exactly 1 minute as given in the question. Why Other Options Are Wrong: Tunnel lengths like 130 metres or 360 metres are too short and would result in a shorter passing time than 60 seconds. Values such as 420 metres or 540 metres do not lead to the exact 60 second figure when recomputed. Only 500 metres produces the correct time with the given speed and train length. Common Pitfalls: Candidates sometimes forget to add the train length to the tunnel length when computing total distance, using only the tunnel length instead. Another frequent mistake is incorrect unit conversion from km/h to metres per second, or misinterpreting 1 minute as 1 hour. Consistent units and a clear understanding of the geometry of the situation are key to solving this type of question. Final Answer: The length of the tunnel is 500 metres.

Question 16

A train is travelling at 48 km/h. It crosses another train, having half of its own length and travelling in the opposite direction at 42 km/h, in 12 seconds. It also passes completely over a railway platform in 45 seconds. What is the length of the …

Accepted Answer

Introduction / Context: This question is structurally similar to a typical two part trains problem, where information about two trains crossing each other is used to deduce the length of one train, and then that length is used with a platform crossing time to find the length of the platform. It tests multi step reasoning, unit conversion, and consistent application of relative speed concepts. Given Data / Assumptions: Speed of the main train = 48 km/h. Speed of the second train = 42 km/h. Length of the second train = half the length of the main train. The trains move in opposite directions and cross each other in 12 seconds. The main train crosses a platform completely in 45 seconds at 48 km/h. We must find the length of the platform in metres. Concept / Approach: Let the length of the main train be L metres, so the other train has length L / 2 metres. First, when two trains run in opposite directions, their relative speed is the sum of their speeds and the total distance covered during crossing is L + L / 2. From this we compute L. Next, when the main train crosses a platform, the distance travelled is L plus the platform length. Using its speed and crossing time, we solve for the platform length. Step-by-Step Solution: Step 1: Convert speeds to metres per second: 48 km/h = 48 * 5 / 18 = 40 / 3 metres per second; 42 km/h = 42 * 5 / 18 = 35 / 3 metres per second.Step 2: Relative speed in opposite directions = 40 / 3 + 35 / 3 = 75 / 3 = 25 metres per second.Step 3: Total distance when they cross each other = L + L / 2 = 3L / 2 metres.Step 4: Time for crossing is 12 seconds, so 3L / 2 = 25 * 12 = 300, giving L = 200 metres.Step 5: Now consider the main train crossing the platform. Let the platform length be P metres.Step 6: Total distance while crossing the platform = L + P = 200 + P.Step 7: Speed of the train = 40 / 3 metres per second, and time taken = 45 seconds, so 200 + P = (40 / 3) * 45 = 600.Step 8: Therefore P = 600 - 200 = 400 metres. Verification / Alternative check: Check the crossing of trains: with L = 200 and the other train 100 metres long, total distance is 300 metres. At relative speed 25 metres per second, time = 300 / 25 = 12 seconds, matching the data.Check platform crossing: train length 200 metres plus platform 400 metres gives 600 metres. At 40 / 3 metres per second, time = 600 / (40 / 3) = 45 seconds, also matching the given information. Why Other Options Are Wrong: Platform lengths such as 320 metres, 360 metres, 480 metres or 500 metres do not satisfy both the crossing and platform conditions when checked with the same speeds and times. Only 400 metres ensures both the 12 second crossing and the 45 second platform passage work out correctly. Common Pitfalls: Common errors include forgetting that the other train has only half the length of the main train or failing to add both train lengths when calculating the crossing distance. Some candidates also confuse km/h with metres per second or mishandle the arithmetic when solving 3L / 2 = 300. Separating the problem clearly into two stages and checking each numerically helps avoid these mistakes. Final Answer: The length of the platform is 400 metres.

Question 17

Along a straight railway track, a jogger runs at 9 km/h and is 240 m ahead of the engine of a train 120 m long that is moving at 45 km/h in the same direction. After how many seconds will the train completely pass the jogger?

Accepted Answer

Introduction / Context: This aptitude question on trains and joggers tests the core concept of relative speed and the extra distance that a train has to travel in order to completely pass a moving person. Such questions are very common in competitive exams because they combine unit conversion, distance–time relations, and careful reading of the phrase "completely pass". Understanding how to model the situation in terms of relative motion is essential for solving similar questions quickly and accurately. Given Data / Assumptions: Speed of the jogger = 9 km/h, moving along the track. Speed of the train = 45 km/h in the same direction. Initial distance between the jogger and the engine of the train = 240 m. Length of the train = 120 m. We assume both jogger and train maintain constant speeds and move in a straight line. Concept / Approach: The main idea is relative speed between two bodies moving in the same direction. When both move in the same direction, the effective speed at which one closes the gap on the other is the difference of their speeds. To completely pass the jogger, the front of the train must first cover the initial gap and then the full length of the train must go past the jogger. Total distance to be covered at relative speed is therefore initial gap plus train length. Step-by-Step Solution: Relative speed in km/h = 45 − 9 = 36 km/h. Convert 36 km/h to m/s: 36 * (5 / 18) = 10 m/s. Initial gap between train engine and jogger = 240 m. Train length = 120 m. Total distance that the train front must cover to completely pass the jogger = 240 m + 120 m = 360 m. Time = distance / relative speed = 360 / 10 = 36 seconds. Verification / Alternative check: We can do a quick check by imagining the frame of reference of the jogger. In that frame, the jogger is at rest and the train approaches at 10 m/s. If the total length that must pass is 360 m, then a time of 36 seconds is reasonable because 10 m/s for 36 seconds covers exactly 360 m. The numbers are consistent and unit conversions are correct. Why Other Options Are Wrong: 18 sec is too small because at 10 m/s the train would only cover 180 m, which is less than the required 360 m. 48 sec is too large and corresponds to a much bigger distance than needed. 72 sec is clearly far too long and would imply a much slower relative speed or a much longer total distance than actually given in the problem. Common Pitfalls: Many learners forget to add both the initial gap and the train length, and instead use only the gap or only the length. Another common mistake is to add the speeds instead of subtracting them when bodies move in the same direction. Some students also forget to convert km/h to m/s correctly, which leads to numerical errors even when the method is right. Final Answer: Thus, the train will completely pass the jogger in 36 sec.

Question 18

A train 170 m long is running at 63 km/h. It can completely cross a bridge in 30 seconds. What is the length of the bridge in metres?

Accepted Answer

Introduction / Context: This question checks understanding of how to calculate distance travelled by a moving train in a given time and how that distance relates to the lengths of both the train and the bridge it crosses. Problems of this type are standard in quantitative aptitude tests, and mastering them helps in quickly solving a variety of train and motion questions. Given Data / Assumptions: Length of the train = 170 m. Speed of the train = 63 km/h. Time taken to completely cross the bridge = 30 seconds. The train runs at constant speed and the bridge is straight. When the train has completely crossed the bridge, its entire length has moved beyond the far end of the bridge. Concept / Approach: When a train crosses a bridge, the distance it travels from the moment its front enters the bridge until its rear leaves the bridge is equal to train length plus bridge length. Using speed and time, we find the total distance covered and then subtract the known train length to get the bridge length. Unit conversion from km/h to m/s is necessary to match with seconds and metres. Step-by-Step Solution: Speed in m/s = 63 * (5 / 18) = 17.5 m/s. Time taken to cross the bridge = 30 s. Total distance covered in this time = speed * time = 17.5 * 30 = 525 m. This total distance equals train length + bridge length. Bridge length = 525 m − 170 m = 355 m. Verification / Alternative check: We can check the calculation by reversing it. If the bridge is 355 m long, total effective length to clear is 170 m + 355 m = 525 m. At 17.5 m/s, time needed is 525 / 17.5 = 30 s, which matches the given information. This confirms that the result is consistent and accurate. Why Other Options Are Wrong: 325 m would give a total distance of 495 m, leading to a time less than 30 s. 312 m yields a total of 482 m, which also contradicts the 30 s duration. 365 m gives a total of 535 m, leading to more than 30 s. None of these match the correct combination of speed and time given in the question. Common Pitfalls: A very common error is to forget that the train must travel its own length plus the bridge length, not only the length of the bridge. Another mistake is to mix units and multiply km/h directly by seconds without converting to m/s, which produces incorrect numerical values. Careful attention to units and the physical meaning of “completely cross” avoids these issues. Final Answer: Therefore, the length of the bridge is 355 m.

Question 19

One train leaves a station at 3:00 p.m. and travels in a straight line at 70 km/h. A second train starts from the same station in the same direction at 4:30 p.m. at 85 km/h. At what time will the faster train catch up with the slower train?

Accepted Answer

Introduction / Context: This problem is a classic example of a chase or catch up scenario involving two trains leaving at different times and travelling at different speeds. Questions like this help develop intuition for relative motion in the same direction, time differences, and distance travelled before one object catches the other. Given Data / Assumptions: Train 1 leaves at 3:00 p.m. with speed 70 km/h. Train 2 leaves the same station at 4:30 p.m. with speed 85 km/h. Both trains move along the same straight track in the same direction. We ignore stops, acceleration, and any other delays. Concept / Approach: First, we find how far the first train travels before the second train even starts. That distance becomes the head start. Then we use the relative speed between the two trains to calculate how long the second train takes to close that distance. Finally, we add this time to the start time of the second train to get the actual meeting time. Step-by-Step Solution: Time gap between departures = 4:30 p.m. − 3:00 p.m. = 1.5 hours. Distance travelled by Train 1 in 1.5 hours = 70 * 1.5 = 105 km. Relative speed when both are moving = 85 − 70 = 15 km/h. Time taken by Train 2 to make up 105 km at 15 km/h = 105 / 15 = 7 hours. Train 2 starts at 4:30 p.m., so meeting time = 4:30 p.m. + 7 hours = 11:30 p.m. Verification / Alternative check: At 11:30 p.m., time since 3:00 p.m. for Train 1 is 8.5 hours, so it covers 70 * 8.5 = 595 km. Time since 4:30 p.m. for Train 2 is 7 hours, so it covers 85 * 7 = 595 km. Both distances match, confirming that the trains meet at 11:30 p.m. Why Other Options Are Wrong: 9:50 p.m., 10:10 p.m., and 10:30 p.m. all correspond to less time for the faster train to catch the slower one, meaning it would not yet have closed the 105 km gap at a relative speed of 15 km/h. Only 11:30 p.m. satisfies the time and distance equations simultaneously. Common Pitfalls: Students often forget to account for the delayed start of the second train and treat both as starting together. Another mistake is to add the speeds instead of taking the difference in a same direction chase problem. Finally, some learners mis-handle the conversion from fractional hours to clock times, so practising this step is important. Final Answer: The faster train will catch up with the slower train at 11:30 p.m.

Question 20

A train is running at 40 km/h and crosses a signal pole in 9 seconds. What is the length of the train in metres?

Accepted Answer

Introduction / Context: This question focuses on a very basic and important concept in train problems: using speed and time to determine distance, which in this case is simply the length of the train. When a train crosses a pole, only its own length has to pass the point, so distance travelled equals train length. Given Data / Assumptions: Speed of the train = 40 km/h. Time taken to cross the pole = 9 seconds. The track is straight and level, and the speed is constant. When crossing a pole, only the length of the train is relevant. Concept / Approach: The fundamental relation is distance = speed * time. However, speed and time must be in compatible units. Since time is given in seconds and the required distance is in metres, we should convert the speed from km/h to m/s before applying the formula. Once we find the distance covered in 9 seconds, that distance is equal to the length of the train. Step-by-Step Solution: Convert speed to m/s: 40 km/h = 40 * (5 / 18) m/s. 40 * (5 / 18) = 200 / 18 ≈ 11.11 m/s. Time = 9 s. Distance travelled in 9 s = 11.11 * 9 ≈ 100 m. Therefore, the length of the train is about 100 m. Verification / Alternative check: We can quickly check by reversing the calculation. If the train is 100 m long and moves at about 11.11 m/s, then the time to cross a pole is 100 / 11.11 ≈ 9 s, which matches the given time. This confirms that the result is consistent. Why Other Options Are Wrong: Lengths like 90 m, 120 m, or 150 m do not match the distance covered by a 40 km/h train in 9 seconds. If the train were 90 m, the time at 11.11 m/s would be about 8.1 s. For 120 m, the time would be about 10.8 s, and for 150 m, about 13.5 s. None of these are equal to 9 seconds. Common Pitfalls: The main mistake is not converting km/h to m/s and directly multiplying by seconds, which leads to unrealistically large distances. Another error is thinking that some extra distance is needed when crossing a pole, but in reality only the train length matters in this type of question. Final Answer: Hence, the length of the train is 100 m.

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