Question 1

Two trains are running on parallel tracks at 60 km/h and 20 km/h respectively in the same direction. The faster train completely passes a man sitting in the slower train in 6 seconds. Approximately what is the length of the faster train in metres?

Accepted Answer

Introduction / Context: This problem extends the concept of relative speed to a situation where a fast train overtakes a man who is sitting in a slower moving train. Since both are moving in the same direction, the effective speed of the fast train relative to the man is the difference of their speeds. The question aims to find the length of the faster train using the time taken to completely pass the man. Given Data / Assumptions: Speed of faster train = 60 km/h. Speed of slower train (where the man is sitting) = 20 km/h. Time taken for faster train to completely pass the man = 6 s. Both trains move in the same direction on parallel tracks with constant speeds. The man is treated as a point on the slower train. Concept / Approach: When two bodies move in the same direction, relative speed = faster speed − slower speed. The distance that must be covered for the faster train to completely pass the man is just the length of the fast train. Once the relative speed is converted to m/s, distance (train length) can be calculated using distance = speed * time. Step-by-Step Solution: Relative speed in km/h = 60 − 20 = 40 km/h. Convert relative speed to m/s: 40 * (5 / 18) ≈ 11.11 m/s. Time taken to pass the man = 6 s. Length of the faster train = relative speed * time = 11.11 * 6 ≈ 66.67 m. Verification / Alternative check: We can check the answer by converting 66.67 m back into time. At a relative speed of about 11.11 m/s, the time taken to cover 66.67 m is 66.67 / 11.11 ≈ 6 s, which matches the data in the question. Therefore, the calculation is consistent and the length is correct to two decimal places. Why Other Options Are Wrong: A length of 54 m or 60 m would result in a passing time smaller than 6 s at the same relative speed. A length of 72 m would require more than 6 s to pass. Since the question specifies 6 s, only a length close to 66.67 m satisfies the given conditions. Common Pitfalls: One frequent error is to add the speeds instead of subtracting them, which is only correct for opposite directions. Another mistake is to think that the slower train length is also involved, but for a man sitting in that train only the length of the faster train needs to pass him. Also, learners sometimes forget to convert km/h to m/s, which leads to numerical errors. Final Answer: Thus, the approximate length of the faster train is 66.67 m.

Question 2

Train K crosses a stationary Train L in 50 seconds and crosses a signal pole in 20 seconds while running at the same constant speed. If the length of Train K is 240 m, what is the length of Train L in metres?

Accepted Answer

Introduction / Context: This question involves a moving train crossing both a stationary pole and a stationary train. It requires understanding that the time taken to cross a pole depends only on the length of the moving train, whereas the time taken to cross another train depends on the sum of their lengths. By combining these two pieces of information, we can find the unknown length of the stationary train. Given Data / Assumptions: Train K length = 240 m. Train K crosses a pole in 20 s. Train K crosses stationary Train L completely in 50 s. Train L is stationary on a parallel track. Train K runs at constant speed during both events. Concept / Approach: First, we use the time taken by Train K to cross a pole to find its speed, since distance equals train length in that case. Then, using the same speed and the time taken to cross Train L, we determine the total distance covered while crossing Train L. That distance equals the sum of the lengths of the two trains, so subtracting the known length of Train K gives us the length of Train L. Step-by-Step Solution: Speed of Train K = distance / time when crossing pole. Speed = 240 m / 20 s = 12 m/s. Time to cross Train L = 50 s. Total distance covered while crossing Train L = speed * time = 12 * 50 = 600 m. Total distance = length of K + length of L. So, 600 = 240 + length of L. Length of Train L = 600 − 240 = 360 m. Verification / Alternative check: We can verify by reversing the logic. If Train L is 360 m long, the combined length is 240 + 360 = 600 m. At 12 m/s, time to cross this distance is 600 / 12 = 50 s, matching the information given. Therefore the answer is consistent. Why Other Options Are Wrong: Lengths like 60 m or 120 m yield total distances 300 m or 360 m, corresponding to crossing times of only 25 s or 30 s, which conflict with the given 50 s. A length of 240 m would make both trains 240 m each, total 480 m, leading to only 40 s. Only 360 m produces the correct 50 s crossing time. Common Pitfalls: Learners sometimes equate the distance travelled while crossing Train L to only the length of Train L, ignoring the length of Train K. Others may try to use km/h without converting units consistently. Remember that for train versus train, the relevant distance is the sum of their lengths, while for train versus pole it is only the train length. Final Answer: Hence, the length of stationary Train L is 360 m.

Question 3

Two trains running in opposite directions cross a man standing on a platform in 27 seconds and 17 seconds respectively, and they cross each other completely in 23 seconds. What is the ratio of their speeds?

Accepted Answer

Introduction / Context: This is a classic relative speed problem where two trains cross a man separately and also cross each other. The question is designed to test the ability to express train lengths in terms of speed and time, form an equation using the crossing time of the trains, and then deduce the ratio of their speeds. Given Data / Assumptions: First train crosses a man in 27 s. Second train crosses a man in 17 s. They cross each other completely in 23 s while moving in opposite directions. Both trains move at constant speeds. The man is stationary on the platform. Concept / Approach: Let speeds of the trains be v1 and v2, and their lengths be l1 and l2. From the times taken to cross the man, we know l1 = v1 * 27 and l2 = v2 * 17 (with consistent units). When the trains cross each other while moving in opposite directions, the relevant relative speed is v1 + v2, and the total distance is l1 + l2. Setting up the equation for the crossing time gives a relationship between v1 and v2, which we can manipulate to obtain the ratio of their speeds. Step-by-Step Solution: Let l1 = length of train 1, l2 = length of train 2. From crossing the man: l1 = v1 * 27 and l2 = v2 * 17. When trains cross each other: time = 23 s. So, (l1 + l2) / (v1 + v2) = 23. Substitute l1 and l2: (27 v1 + 17 v2) / (v1 + v2) = 23. 27 v1 + 17 v2 = 23 v1 + 23 v2. 4 v1 = 6 v2, so v1 / v2 = 6 / 4 = 3 / 2. Verification / Alternative check: Once the ratio v1 : v2 = 3 : 2 is known, we can assign simple values, for example v1 = 3k and v2 = 2k, and check the equation. Then l1 = 27 * 3k and l2 = 17 * 2k. Substituting back into the expression for crossing each other gives exactly 23 s, which confirms that the ratio 3 : 2 is correct. Why Other Options Are Wrong: Ratios like 1 : 2, 3 : 1, or 4 : 7 do not satisfy the equation (27 v1 + 17 v2) / (v1 + v2) = 23. If we plug those ratios into the relationship, the left side and right side will not match, so they are incompatible with the given times. Common Pitfalls: A common error is to assume that the ratio of speeds equals the inverse ratio of times taken to cross the man (27 and 17), which is not directly true here because the trains have different lengths as well. Another mistake is to forget that, for trains crossing each other, the relative speed is the sum of their speeds, not the difference. Final Answer: The ratio of the speeds of the two trains is 3 : 2.

Question 4

A train takes 10 seconds to cross a pole and twice this time to cross a platform of length 200 m. What is the length of the train in metres?

Accepted Answer

Introduction / Context: This problem combines two related situations: a train crossing a pole and the same train crossing a platform. It tests basic understanding of how distance, speed, and time interact, and how the effective distance changes depending on whether the train is passing a point object or an extended object like a platform. Given Data / Assumptions: Time taken to cross a pole = 10 s. Time taken to cross a platform = 20 s (twice 10 s). Length of the platform = 200 m. The train moves with constant speed in both cases. The track is straight and there are no stops or changes in speed. Concept / Approach: When a train crosses a pole, the distance covered is just the length of the train. When it crosses a platform, the distance covered from the instant the front of the train reaches the platform until the rear leaves it is the sum of the train length and platform length. Since the time for the second scenario is double the first, we can relate the two distances using the same train speed. Step-by-Step Solution: Let L be the length of the train in metres. When crossing a pole, distance = L and time = 10 s. Speed of the train = L / 10 m/s. When crossing the platform, distance = L + 200 m and time = 20 s. Speed must be the same, so (L + 200) / 20 = L / 10. Multiply both sides by 20: L + 200 = 2L. Rearrange: 2L − L = 200, so L = 200 m. Verification / Alternative check: If the train is 200 m long, then its speed is 200 / 10 = 20 m/s when crossing the pole. When crossing the platform, distance = 200 + 200 = 400 m. At 20 m/s, time = 400 / 20 = 20 s, which matches the given condition of double the original time. Why Other Options Are Wrong: If the train were 180 m long, the crossing time for the platform would not be exactly twice 10 s. Similar mismatches occur for lengths 190 m and 210 m, because they do not satisfy the equation (L + 200) / 20 = L / 10. Only 200 m gives a consistent speed across both scenarios. Common Pitfalls: Some students mistakenly think that platform length does not matter or they confuse which distance corresponds to which time. Others forget that the same train speed applies in both situations, which is crucial for equating the two distance over time expressions. Carefully setting up the equation avoids these errors. Final Answer: The length of the train is 200 m.

Question 5

Two trains are running at 40 km/h and 20 km/h respectively in the same direction on parallel tracks. The faster train completely passes a man sitting in the slower train in 5 seconds. What is the length of the faster train in metres?

Accepted Answer

Introduction / Context: This question again uses the concept of relative speed for two trains moving in the same direction. The goal is to use the time taken for the faster train to overtake a man sitting in the slower train to compute the length of the faster train. The presence of mixed fraction options hints that the exact answer is a repeating decimal that is more neatly expressed as a fraction. Given Data / Assumptions: Speed of faster train = 40 km/h. Speed of slower train (where the man is sitting) = 20 km/h. Time taken for the faster train to pass the man completely = 5 s. Both trains travel in the same direction on parallel tracks with constant speeds. The man is treated as a point object. Concept / Approach: The effective speed of the faster train relative to the man is the difference of their speeds, since they move in the same direction. Once we convert this relative speed to m/s, the length of the faster train is simply the product of the relative speed and the overtaking time. The result will be a fraction, which we can convert into a mixed fraction format to match the options. Step-by-Step Solution: Relative speed in km/h = 40 − 20 = 20 km/h. Convert to m/s: 20 * (5 / 18) = 100 / 18 = 50 / 9 m/s. Time to pass the man = 5 s. Length of faster train = relative speed * time = (50 / 9) * 5 = 250 / 9 m. 250 / 9 = 27 remainder 7, so 27 7/9 m. Verification / Alternative check: Check by reversing: at relative speed 50 / 9 m/s, a length of 250 / 9 m will take (250 / 9) / (50 / 9) = 5 s to pass, which matches the given time exactly. This confirms that the calculation and fractional form are correct. Why Other Options Are Wrong: Values like 25 8/7 m, 21 1/4 m or 22 m do not equal 250 / 9 m when converted to improper fractions. Using those lengths with the same relative speed would give times different from 5 s, so they cannot satisfy the given condition of the question. Common Pitfalls: A major pitfall is using the sum of the two speeds instead of the difference, which is only valid for opposite directions. Another error is to round the fraction 250 / 9 too early, which can lead to confusion when comparing with fractional options. Keeping the exact fraction until the end makes it easy to recognize the correct mixed number. Final Answer: The length of the faster train is 27 7/9 m.

Question 6

Two goods trains, each 520 m long, are running in opposite directions on parallel tracks at speeds of 42 km/h and 36 km/h respectively. How many seconds will they take to completely cross each other?

Accepted Answer

Introduction / Context: This question deals with two trains of equal length running in opposite directions. It checks understanding of how to compute relative speed when trains move towards each other and how to apply the distance equals speed times time relationship when the distance to be covered is the sum of both train lengths. Given Data / Assumptions: Length of each goods train = 520 m. Speed of first train = 42 km/h. Speed of second train = 36 km/h. They move in opposite directions on parallel tracks. We want the time for them to completely cross each other. Concept / Approach: When two objects move towards each other, the relative speed is the sum of their individual speeds. To cross each other completely, the front of each train must travel past the entire length of the other train, so the total distance involved is the sum of both lengths. Once we convert the relative speed to m/s, we can compute the time as distance divided by relative speed. Step-by-Step Solution: Total distance to be covered while crossing = 520 m + 520 m = 1040 m. Relative speed in km/h = 42 + 36 = 78 km/h. Convert relative speed to m/s: 78 * (5 / 18) = 390 / 18 ≈ 21.67 m/s. Time taken to cross each other = distance / relative speed = 1040 / 21.67 ≈ 48 s. Verification / Alternative check: We can keep the values exact: 78 km/h = 78 * 5 / 18 = 390 / 18 = 65 / 3 m/s. Then time = 1040 / (65 / 3) = 1040 * 3 / 65 = 3120 / 65 = 48 s exactly. This confirms that 48 seconds is not just an approximation but the exact value based on the given data. Why Other Options Are Wrong: Times such as 60 s, 45 s, or 34 s do not correspond to the correct ratio of distance to relative speed. If we multiply these times by the relative speed 65 / 3 m/s, the resulting distances are not equal to 1040 m, so they cannot represent the correct crossing time for the given train lengths. Common Pitfalls: Learners sometimes forget to add both train lengths and use only one length, which halves the required distance and gives a wrong time. Another error is to subtract the speeds instead of adding them, which is appropriate only when trains move in the same direction. Additionally, incorrect conversion between km/h and m/s can significantly change the numerical result. Final Answer: The two goods trains will completely cross each other in 48 sec.

Question 7

Trains X and Y leave station A at 6:30 a.m. and 7:40 a.m. respectively and travel in the same direction at 30 km/h and 40 km/h. How many kilometres from station A will the two trains meet?

Accepted Answer

Introduction / Context: This is another chase problem involving two trains leaving the same station at different times and different speeds. It checks the ability to compute head start distance, relative speed, and then translate the time of meeting into distance from the starting point. Given Data / Assumptions: Train X leaves station A at 6:30 a.m. at 30 km/h. Train Y leaves station A at 7:40 a.m. at 40 km/h. Both trains travel in the same direction on the same route. We need the distance from station A where they meet. Speeds remain constant and trains do not stop. Concept / Approach: First, compute how long Train X travels before Train Y starts, and convert that time into a head start distance. Then find the relative speed between the trains, treating it as a chase. Using relative speed, determine how long Train Y takes to catch Train X. Finally, multiply the total time Train X has been travelling by its speed to get the distance from station A where they meet. Step-by-Step Solution: Time gap between departures = 7:40 a.m. − 6:30 a.m. = 1 hour 10 minutes = 7/6 hours. Head start distance of Train X = 30 * (7/6) = 35 km. Relative speed = 40 − 30 = 10 km/h. Time taken by Train Y to close the 35 km gap = 35 / 10 = 3.5 hours. Total time Train X travels until meeting = 7/6 + 3.5 = 7/6 + 7/2 = 14/3 hours. Distance from A at meeting point = 30 * (14/3) = 140 km. Verification / Alternative check: We can also compute from Train Y perspective. Train Y travels for 3.5 hours at 40 km/h, covering 140 km. This matches the distance we obtained from the perspective of Train X, so both views are consistent. Therefore the meeting point is 140 km from station A. Why Other Options Are Wrong: Distances like 96 km, 120 km, or 142 km do not satisfy the time and speed relationships. For example, 120 km would require different travel times from each perspective that conflict with the calculated head start and relative speed, so they cannot be correct meeting distances for the data given. Common Pitfalls: A common mistake is to ignore the delayed start of Train Y and assume both start together. Another frequent issue is mis-converting 1 hour 10 minutes into fractional hours. Handling time carefully and using relative speed correctly ensures an accurate solution. Final Answer: The two trains will meet at a point 140 km from station A.

Question 8

A train 125 m long passes a man who is running at 5 km/h in the same direction as the train in 10 seconds. What is the speed of the train in km/h?

Accepted Answer

Introduction / Context: This question tests relative speed concepts when a train overtakes a man moving in the same direction. The train length and overtaking time are given, and the objective is to find the speed of the train in km/h. Such questions are common in aptitude exams and help develop quick reasoning about motion along a straight line. Given Data / Assumptions: Length of the train = 125 m. Speed of the man = 5 km/h. Time taken by the train to pass the man completely = 10 s. The man and the train move in the same direction. Speeds are constant and motion is along a straight track. Concept / Approach: The train has to cover its own length relative to the man in order to overtake him fully. Therefore, the relevant speed is the relative speed between the train and the man, which is the difference between their speeds in the same direction. We can first compute the relative speed in m/s using distance and time, then convert that into km/h and finally add the man's speed to get the train's speed. Step-by-Step Solution: Relative distance to be covered = length of train = 125 m. Time to overtake = 10 s. Relative speed in m/s = distance / time = 125 / 10 = 12.5 m/s. Convert relative speed to km/h: 12.5 * (18 / 5) = 12.5 * 3.6 = 45 km/h. Let speed of train = V km/h. Then relative speed = V − 5. So, V − 5 = 45 which gives V = 50 km/h. Verification / Alternative check: Check by forward calculation: If the train moves at 50 km/h and the man at 5 km/h, relative speed = 45 km/h. In m/s, this is 45 * 5 / 18 = 12.5 m/s. Time to cover the train length 125 m = 125 / 12.5 = 10 s, which agrees with the given time. Why Other Options Are Wrong: If the train speed were 45 km/h, relative speed would be only 40 km/h, leading to a different overtaking time. Similarly, speeds 25 km/h or 30 km/h are much too slow and cannot allow the train to pass a man moving at 5 km/h in 10 seconds. Only 50 km/h satisfies both the relative speed and time conditions. Common Pitfalls: Some learners mistakenly add the speeds instead of subtracting them in same direction motion. Others compute relative speed correctly but forget to add the man's speed back to obtain the train's speed. Careful interpretation of the phrase "same direction" and a two step approach for relative and actual speeds help avoid these mistakes. Final Answer: The speed of the train is 50 km/h.

Question 9

A train 110 m long is running at 60 km/h. In how many seconds will it pass a man who is running at 6 km/h in the opposite direction to that of the train?

Accepted Answer

Introduction / Context: This problem illustrates the situation where a train and a man move in opposite directions. The focus is on using relative speed when motions are towards each other and finding the time required for the train to pass the man completely. It is a standard type of question in aptitude examinations on trains and relative motion. Given Data / Assumptions: Length of the train = 110 m. Speed of the train = 60 km/h. Speed of the man = 6 km/h. The man runs in the direction opposite to the motion of the train. We assume constant speeds and straight line motion. Concept / Approach: When two objects move in opposite directions, their relative speed is the sum of their individual speeds. To pass the man completely, the train must cover a relative distance equal to its own length. Once we obtain the relative speed in m/s, we can use time = distance / speed to compute the passing time. Step-by-Step Solution: Relative speed in km/h = 60 + 6 = 66 km/h. Convert relative speed to m/s: 66 * (5 / 18) = 330 / 18 = 55 / 3 m/s. Distance to be covered relative to the man = length of train = 110 m. Time taken = distance / relative speed = 110 / (55 / 3) = 110 * 3 / 55. Simplify: 110 * 3 / 55 = 6 s. Verification / Alternative check: We can reason intuitively. A relative speed of 66 km/h is roughly 18.33 m/s. At that rate, covering 110 m should take about 110 / 18.33 ≈ 6 s, which aligns with the exact calculation. This gives confidence that both unit conversion and arithmetic are correct. Why Other Options Are Wrong: Times of 5 s, 7 s, or 10 s do not satisfy the relation time = 110 / (55 / 3). Using these times with the correct relative speed would not yield the correct train length. Only 6 s matches both the given data and the fundamental distance, speed, and time equations. Common Pitfalls: A common mistake is to subtract the speeds instead of adding them for opposite direction motion. Another error is mixing units by not converting km/h to m/s when using metres and seconds for distance and time. Following a consistent unit strategy and remembering that opposite direction means speeds add avoids these errors. Final Answer: The train will pass the man completely in 6 sec.

Question 10

Two superfast trains have lengths 140 m and 160 m respectively. They run towards each other on parallel tracks at 60 km/h and 80 km/h. In how many seconds will they completely cross each other?

Accepted Answer

Introduction / Context: This question is another example of trains running in opposite directions. It asks for the time needed for two superfast trains of different lengths to completely cross each other. The problem combines relative speed, the total effective distance, and correct unit handling. Given Data / Assumptions: Length of first train = 140 m. Length of second train = 160 m. Speed of first train = 60 km/h. Speed of second train = 80 km/h. They run on parallel tracks in opposite directions. Concept / Approach: When two trains run towards each other, the relative speed is the sum of their speeds. The total distance that needs to be covered for each train to completely clear the other is the sum of their lengths. After converting the relative speed into m/s, we can find the crossing time as distance divided by relative speed. Step-by-Step Solution: Total distance for complete crossing = 140 m + 160 m = 300 m. Relative speed in km/h = 60 + 80 = 140 km/h. Convert to m/s: 140 * (5 / 18) = 700 / 18 ≈ 38.89 m/s. Time taken to cross each other = distance / relative speed = 300 / 38.89 ≈ 7.71 s. Verification / Alternative check: Using exact fractions, 140 km/h = 140 * 5 / 18 = 700 / 18 m/s. Then time = 300 / (700 / 18) = 300 * 18 / 700 = 5400 / 700 = 54 / 7 ≈ 7.714 s. Rounded to two decimal places, this is 7.71 s, which matches the answer option. Why Other Options Are Wrong: Times such as 9.36 s, 8.45 s, or 10.48 s would correspond to different effective speeds or distances and do not satisfy the equation time = 300 / (700 / 18). If we multiply any of these times by the relative speed, we do not get a distance of 300 m, so they cannot represent the correct crossing time. Common Pitfalls: Mistakes often include subtracting the speeds instead of adding them and forgetting to add the lengths of both trains when calculating the distance. Also, some students mis-handle decimal approximations. Keeping exact fractional values as long as possible helps to avoid rounding errors and identify the correct option. Final Answer: The two superfast trains will completely cross each other in approximately 7.71 sec.

Question 11

Agra and Delhi are 160 km apart. A train leaves Agra for Delhi and at the same time another train leaves Delhi for Agra. The two trains meet 5 hours after they start. If the train from Agra to Delhi travels 6 km/h faster than the other train, what i…

Accepted Answer

Introduction / Context: This is a typical meeting point problem involving two trains starting from opposite cities and moving towards each other. The question tests understanding of combined speed, distance covered before meeting, and how to use a small difference between speeds to solve for individual speeds. Given Data / Assumptions: Distance between Agra and Delhi = 160 km. The trains start at the same time from opposite cities. They meet after 5 hours. Speed of train from Agra to Delhi is 6 km/h more than the speed of the other train. Both trains travel at constant speeds. Concept / Approach: Let the speed of the slower train be v km/h. Then the speed of the faster train is v + 6 km/h. When two trains move towards each other, the total distance covered until they meet equals the sum of the distances each train has travelled. This is also equal to (v + (v + 6)) multiplied by the time before meeting. By equating this product to the known distance, we can solve for v and then determine the faster train speed. Step-by-Step Solution: Let speed of slower train = v km/h. Then speed of faster train = v + 6 km/h. Time before meeting = 5 hours. Total distance covered = distance from Agra to Delhi = 160 km. So, (v + v + 6) * 5 = 160. (2v + 6) * 5 = 160 → 2v + 6 = 32 → 2v = 26 → v = 13 km/h. Speed of faster train = v + 6 = 13 + 6 = 19 km/h. Verification / Alternative check: At 13 km/h, the slower train travels 13 * 5 = 65 km in 5 hours. At 19 km/h, the faster train travels 19 * 5 = 95 km. The sum is 65 + 95 = 160 km, which equals the distance between the cities. Thus both the meeting time and the distance are consistent, confirming the answer. Why Other Options Are Wrong: Values such as 7 km/h, 9 km/h, or 13 km/h for the faster train do not satisfy the equation (v + (v + 6)) * 5 = 160. In fact, 13 km/h is the slower train speed, not the faster one. Only 19 km/h produces the correct combined distance of 160 km in 5 hours. Common Pitfalls: Students sometimes misinterpret which train is faster or forget to multiply by the time to get total distance. Others may assume equal speeds, which contradicts the given difference of 6 km/h. Proper variable assignment and careful equation setup help avoid these errors. Final Answer: The speed of the faster train is 19 km/h.

Question 12

An engine of a train of length 1000 m is moving along a straight track at 10 m/s. A bird starts from the engine and flies along the train to the rear end at a speed of x m/s relative to the ground, then immediately flies back from the rear end to th…

Accepted Answer

Introduction / Context: This is a relatively advanced relative motion problem involving a bird flying back and forth along a moving train. The length of the train, speed of the train, and total time for the two legs of the bird's journey are given. The question checks deeper understanding of how relative speeds work when the reference points themselves are in motion. Given Data / Assumptions: Length of the train = 1000 m. Speed of the train = 10 m/s, moving forward along the track. Bird flies from engine to rear end at speed x m/s relative to the ground. Bird then flies back from rear end to engine at speed 2x m/s relative to the ground. Total time for both legs of the bird's journey = 187.5 s. All motions are along the straight line of the train. Concept / Approach: The key idea is to compute the relative speed between the bird and the part of the train it is moving towards on each leg. When the bird flies from the front towards the rear, it effectively moves backward while the rear is moving forward. When it comes back from rear to front, both bird and engine move forward, but at different speeds. Distances between the bird and the target ends are 1000 m for each leg in the ground frame. By setting up time expressions for each leg using these relative speeds and summing them, we can solve for x and then express x and 2x in km/h. Step-by-Step Solution: Let train speed = 10 m/s. First leg (engine to rear): bird moves backward relative to ground at speed x, rear moves forward at 10 m/s. Relative speed between bird and rear end = x + 10 m/s. Time for first leg t1 = 1000 / (x + 10). Second leg (rear to engine): bird moves forward at 2x m/s, engine moves forward at 10 m/s. Relative speed between bird and engine = 2x − 10 m/s (2x must be greater than 10). Time for second leg t2 = 1000 / (2x − 10). Total time t1 + t2 = 187.5 s gives 1000 / (x + 10) + 1000 / (2x − 10) = 187.5. Solving this equation yields x ≈ 8.7284 m/s. Then 2x ≈ 17.4568 m/s. Convert to km/h: x ≈ 8.7284 * 3.6 ≈ 31.4208 km/h and 2x ≈ 17.4568 * 3.6 ≈ 62.8416 km/h. Verification / Alternative check: Substitute x ≈ 8.7284 into t1 and t2. Then t1 = 1000 / (10 + 8.7284) and t2 = 1000 / (2 * 8.7284 − 10). Adding these numerically gives approximately 187.5 s, confirming that the values of x and 2x fit the total time constraint very closely. Hence the corresponding option with 31.4208 and 62.8416 km/h is correct. Why Other Options Are Wrong: If we pick 21.4 and 42.8, 25.2 and 50.4, or 33.12 and 66.24 km/h and convert them back to m/s, the resulting values of x will not satisfy the equation 1000 / (x + 10) + 1000 / (2x − 10) = 187.5. Their computed total times will differ significantly from 187.5 s, so they cannot be correct solutions. Common Pitfalls: Typical mistakes include assuming that the train is stationary, which ignores relative motion, or incorrectly setting relative speeds (for example, subtracting instead of adding in the first leg). Some also forget that distances between bird and train ends are 1000 m for both legs in the ground frame. Careful diagramming of each leg and consistent use of relative speed formulas are essential. Final Answer: Thus, the correct pair of speeds is 31.4208 and 62.8416 km/h.

Question 13

Two passenger trains start at the same time from two different stations and move towards each other at 16 km/h and 21 km/h respectively. When they meet, one train has travelled 60 km more than the other. What is the distance between the two stations?

Accepted Answer

Introduction / Context: This is a distance and relative motion problem involving two trains travelling towards each other at different speeds. The extra piece of information that one train travels 60 km more than the other at the meeting point allows us to set up a simple linear equation to find their travel time and then compute the total distance between stations. Given Data / Assumptions: Speed of first train = 16 km/h. Speed of second train = 21 km/h. They start at the same time from two different stations. They move towards each other on a straight route. At the meeting point, one train has travelled 60 km more than the other. Concept / Approach: If both trains travel for the same time t hours until they meet, distances covered are 16t and 21t. The faster train naturally covers more distance. According to the problem, the difference between these distances is 60 km. So we can write 21t − 16t = 60, solve for t, and then compute the total distance between stations as the sum of the distances travelled by both trains. Step-by-Step Solution: Let t = time in hours until the trains meet. Distance travelled by slower train = 16t. Distance travelled by faster train = 21t. Given that the faster train travels 60 km more: 21t − 16t = 60. 5t = 60 → t = 60 / 5 = 12 hours. Distance by slower train = 16 * 12 = 192 km. Distance by faster train = 21 * 12 = 252 km. Total distance between stations = 192 + 252 = 444 km. Verification / Alternative check: We can verify that the difference between 252 km and 192 km is indeed 60 km as required. Also, the total distance 444 km divided by the combined speed (16 + 21 = 37 km/h) gives time 444 / 37 = 12 hours, which matches our value of t. Both checks confirm that the calculations are consistent. Why Other Options Are Wrong: Distances such as 387 km, 242 km, or 145 km do not satisfy both the total distance and the 60 km difference condition when combined with speeds of 16 km/h and 21 km/h. Using these distances with the given speeds would produce either a different time or a different difference in distances, so they cannot be correct. Common Pitfalls: A common error is to misinterpret the 60 km difference as the total distance instead of the difference, or to forget that both trains travel for the same time t. Another mistake is to try to apply relative speed directly without using the extra difference information correctly. Setting up a clear equation for distance difference is the safest method. Final Answer: The distance between the two stations is 444 km.

Question 14

Two trains of equal length are running at 60 km/h and 40 km/h. They take 50 seconds to cross each other when running in the same direction. How many seconds will they take to cross each other when running in opposite directions?

Accepted Answer

Introduction / Context: This question links two related scenarios: trains running in the same direction and in opposite directions. The trains have equal lengths and we know the time they take to cross each other in one case. Using that, we can deduce the train length and then find the time to cross in the other case. It illustrates how relative speed changes with direction. Given Data / Assumptions: Speed of faster train = 60 km/h. Speed of slower train = 40 km/h. Trains have equal lengths. Time to cross each other in same direction = 50 s. We need time to cross each other in opposite directions. Concept / Approach: When trains move in the same direction, relative speed is the difference between their speeds. The distance for complete crossing is the sum of both train lengths. When they move in opposite directions, the distance is the same, but the relative speed is the sum of their speeds. Using the time in the same direction case, we can compute the combined length and then use it with the new relative speed to get the required time. Step-by-Step Solution: Relative speed in same direction = 60 − 40 = 20 km/h. Convert to m/s: 20 * (5 / 18) = 50 / 9 m/s. Let length of each train be L m, so total length = 2L. Time to cross in same direction = 50 s, so 2L = (50 / 9) * 50 = 2500 / 9 m. Thus, 2L = 2500 / 9 ⇒ L = 1250 / 9 m (not needed explicitly later). Now in opposite directions, relative speed = 60 + 40 = 100 km/h = 100 * (5 / 18) = 250 / 9 m/s. Time to cross in opposite directions = total length / new relative speed = (2500 / 9) / (250 / 9) = 10 s. Verification / Alternative check: We can also reason that time in opposite directions must be smaller by a factor equal to the ratio of relative speeds. Relative speed same direction is 20 km/h; opposite direction is 100 km/h. Factor = 20/100 = 1/5. So time opposite direction = 50 * (1/5) = 10 s. This matches our detailed calculation. Why Other Options Are Wrong: Values 11 s, 12 s, or 8 s do not follow from the ratio of relative speeds or the distance found from the first case. Only 10 s gives the correct proportional reduction in time when relative speed increases by a factor of 5 from 20 km/h to 100 km/h. Common Pitfalls: Some students incorrectly keep using the same relative speed in both cases, or forget to convert km/h to m/s consistently. Others may mistakenly treat the train lengths as changing. Remember that only the relative speed changes when the direction arrangement changes; the train lengths and total crossing distance remain the same. Final Answer: They will take 10 sec to cross each other when running in opposite directions.

Question 15

Two trains of equal length are running on parallel tracks in the same direction at 36 km/h and 26 km/h. The faster train passes the slower train completely in 36 seconds. What is the length of each train in metres?

Accepted Answer

Introduction / Context: Here we have two trains of equal length moving in the same direction at different speeds. The question asks for the length of each train, given the time taken by the faster train to completely overtake the slower one. This problem reinforces the concept of relative speed and the total distance that must be covered in an overtaking situation. Given Data / Assumptions: Speed of faster train = 36 km/h. Speed of slower train = 26 km/h. Time taken to pass completely = 36 s. Lengths of the trains are equal. Trains move on parallel tracks with constant speeds. Concept / Approach: When one train overtakes another moving in the same direction, the relevant speed is the difference of their speeds. The distance that must be covered is the sum of both train lengths because the faster train needs to completely clear the slower one. Once the relative speed is converted to m/s, the total length is speed times time, and dividing by two gives the length of each train. Step-by-Step Solution: Relative speed in km/h = 36 − 26 = 10 km/h. Convert to m/s: 10 * (5 / 18) = 50 / 18 = 25 / 9 m/s. Let L be the length of each train, so total distance for overtaking = 2L. Time for overtaking = 36 s. So, 2L = relative speed * time = (25 / 9) * 36. 2L = (25 / 9) * 36 = 25 * 4 = 100 m. Therefore, L = 100 / 2 = 50 m. Verification / Alternative check: Check by substitution: If each train is 50 m, total length = 100 m. At relative speed 25 / 9 m/s, time = 100 / (25 / 9) = 100 * 9 / 25 = 36 s, which matches the given time. This consistency confirms the correctness of the length. Why Other Options Are Wrong: Lengths like 24 m, 28 m, or 54 m would produce total lengths of 48 m, 56 m, or 108 m, which do not give a crossing time of 36 s at the given relative speed. Only 50 m leads to the correct time based on the speed difference and overtaking time. Common Pitfalls: Learners may forget that for overtaking, the total distance is the sum of both train lengths, not only one length. Another mistake is converting km/h to m/s incorrectly or using the sum of speeds instead of the difference for same direction motion. Final Answer: The length of each train is 50 m.

Question 16

Two trains of lengths 400 m and 600 m are running in the same direction on parallel tracks. The faster train completely overtakes the slower train in 180 seconds. If the speed of the slower train is 48 km/h, what is the speed of the faster train?

Accepted Answer

Introduction / Context: This question covers overtaking between two trains of different lengths moving in the same direction. One train is faster, and the time needed to overtake completely is given, along with the speed of the slower train. The task is to determine the speed of the faster train, using the total distance involved and relative speed. Given Data / Assumptions: Length of faster train = 400 m. Length of slower train = 600 m. Time taken for the faster train to completely overtake the slower train = 180 s. Speed of slower train = 48 km/h. Both trains move in the same direction on parallel tracks at constant speeds. Concept / Approach: In an overtaking scenario, the distance that the faster train must cover relative to the slower train is the sum of both train lengths. The relative speed in the same direction is the difference between their speeds. We can first compute the relative speed in m/s from the given distance and time, convert it to km/h, and then add it to the speed of the slower train to find the speed of the faster train. Step-by-Step Solution: Total distance for complete overtaking = 400 m + 600 m = 1000 m. Time taken = 180 s. Relative speed in m/s = distance / time = 1000 / 180 = 50 / 9 m/s. Convert relative speed to km/h: (50 / 9) * (18 / 5) = (50 * 18) / (9 * 5) = 1000 / 45 = 20 km/h. Let speed of faster train = V km/h. Relative speed = V − 48 = 20, so V = 48 + 20 = 68 km/h. Verification / Alternative check: Check by direct calculation. If slower train speed is 48 km/h and faster train speed is 68 km/h, relative speed = 20 km/h. In m/s, 20 km/h = 20 * 5 / 18 = 50 / 9 m/s. Time needed to cover 1000 m at 50 / 9 m/s is 1000 / (50 / 9) = 1000 * 9 / 50 = 180 s, which matches the given time. Why Other Options Are Wrong: If the faster train moved at 50 km/h or 52 km/h, relative speed would be too small and the overtaking would take much longer than 180 s. If it moved at 76 km/h, relative speed would be 28 km/h, making the overtaking time shorter than 180 s. Only 68 km/h leads to the correct overtaking time for the given distance. Common Pitfalls: Errors often arise from forgetting that total distance is the sum of both train lengths, incorrectly converting km/h to m/s, or adding speeds instead of subtracting them in a same direction overtaking scenario. Carefully identifying relative speed and distance ensures a correct solution. Final Answer: The speed of the faster train is 68 km/h.

Question 17

Two trains are moving in the same direction at speeds of 72 km/h and 36 km/h. The faster train completely crosses a girl who is sitting at a window seat in the slower train in 32 seconds. What is the length of the faster train in metres?

Accepted Answer

Introduction / Context: In many aptitude exams, train problems test the idea of relative speed when two objects move in the same direction or opposite directions. In this question, two trains move in the same direction, and the faster train overtakes a girl sitting in the slower train. We are asked to find the length of the faster train using the time taken to completely pass the girl and the relative speed between the trains. Given Data / Assumptions: Speed of the faster train = 72 km/h. Speed of the slower train = 36 km/h. The girl is sitting in the slower train, treated as a point object. Time taken by the faster train to completely cross the girl = 32 seconds. We assume uniform speeds and straight tracks. Concept / Approach: When two bodies move in the same direction, the effective or relative speed is the difference between their individual speeds. Here, the faster train must cover its own length relative to the girl in the slower train. Once we calculate the relative speed in metres per second, we multiply it by the time taken to cross in order to get the length of the faster train. Step-by-Step Solution: Step 1: Relative speed in km/h = 72 - 36 = 36 km/h. Step 2: Convert this into m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed in m/s = 36 * 5/18 = 10 m/s. Step 4: Time taken to cross the girl = 32 s. Step 5: Length of the faster train = relative speed * time = 10 * 32 = 320 m. Verification / Alternative check: We can verify the calculation quickly. If the faster train is 320 m long and the relative speed is 10 m/s, then time to cross the girl = distance / speed = 320 / 10 = 32 s, which matches the given time. The use of relative speed is correct because both trains are moving in the same direction and the girl shares the motion of the slower train. Why Other Options Are Wrong: 170 m, 100 m, and 270 m are all smaller than the correct length and would give times less than 32 seconds at 10 m/s. For example, 270 m at 10 m/s would give 27 s. The distractor 220 m also leads to 22 s, which does not match the given crossing time. Therefore, none of these satisfy the condition of 32 seconds. Common Pitfalls: A common mistake is to add the two speeds instead of subtracting them when the motion is in the same direction. Another error is to forget to convert from km/h to m/s before multiplying by time in seconds. Some learners also incorrectly treat the entire slower train length, but only the length of the faster train matters when overtaking a single point like a sitting passenger. Final Answer: The length of the faster train is 320 m.

Question 18

Train K crosses a pole in 30 seconds and train L crosses the same pole in 1 minute 20 seconds. The length of train K is three fourths of the length of train L. What is the ratio of the speed of train K to the speed of train L?

Accepted Answer

Introduction / Context: This problem involves two trains crossing the same pole in different times and having a known relationship between their lengths. Such questions test understanding of the relation speed = distance / time and how proportional reasoning can be used when actual lengths are not given but ratios are. We are asked to find the ratio of speeds of the two trains based on their crossing times and relative lengths. Given Data / Assumptions: Train K crosses a pole in 30 seconds. Train L crosses the same pole in 1 minute 20 seconds = 80 seconds. Length of train K = three fourths of the length of train L. Both trains move at constant speeds on straight tracks. When a train crosses a pole, the distance covered is equal to its own length. Concept / Approach: Speed is equal to distance divided by time. When a train crosses a pole, the distance is simply the length of the train. Therefore, the speed of each train is proportional to its length divided by its crossing time. With the ratio of lengths known and times given, we can find the ratio of speeds without needing the actual lengths in metres. Step-by-Step Solution: Step 1: Let the length of train L be L metres. Step 2: Then length of train K = (3/4) * L metres. Step 3: Speed of train K = distance / time = (3/4 * L) / 30. Step 4: Speed of train L = L / 80. Step 5: Ratio of speeds K : L = [(3/4 * L) / 30] : [L / 80]. Step 6: Cancel L and simplify: (3/4) / 30 divided by 1 / 80 = (3/4) * (80 / 30) = 3 * 80 / (4 * 30) = 240 / 120 = 2 / 1. Step 7: Therefore, the ratio of speeds is 2 : 1. Verification / Alternative check: We can assign a convenient value, for example L = 80 m. Then train L speed = 80 / 80 = 1 m/s. Train K length = 3/4 * 80 = 60 m. Its speed = 60 / 30 = 2 m/s. So the ratio of speeds is 2 : 1, which confirms the result. The absolute values were not necessary since ratios suffice. Why Other Options Are Wrong: Ratios like 1 : 3, 3 : 1, and 1 : 2 contradict the fact that train K is shorter but crosses the pole in much less time, so it must be faster, not slower. The distractor 3 : 2 does not match the simplified exact ratio obtained by algebraic cancellation. Common Pitfalls: A common mistake is to compare times directly without including the effect of different lengths. Another pitfall is to confuse three fourths with four thirds, which would reverse the ratio. Students may also forget to convert 1 minute 20 seconds to 80 seconds correctly, which can distort the final ratio. Final Answer: The ratio of the speed of train K to the speed of train L is 2 : 1.

Question 19

A train 125 metres long passes a man who is running at 4 km/h in the same direction as the train. If the train takes 10 seconds to completely overtake the man, what is the speed of the train in km/h?

Accepted Answer

Introduction / Context: This question combines relative speed and unit conversion from km/h to m/s, both of which are frequent in quantitative aptitude problems involving trains and moving persons. The train and the man are moving in the same direction, so the train only gains on the man at the relative speed. The time taken to completely cross the man depends on the train length and this relative speed. Given Data / Assumptions: Length of the train = 125 metres. Speed of the man = 4 km/h. The man and the train move in the same direction. Time taken by the train to pass the man completely = 10 seconds. Speeds are uniform and motion is along a straight track. Concept / Approach: When two bodies move in the same direction, the effective speed that determines how fast one overtakes the other is the difference of their speeds. Here, the distance covered relative to the man is the full length of the train. So, relative speed = train speed minus man speed. We first find relative speed in metres per second from the length and time, then convert the final train speed to km/h. Step-by-Step Solution: Step 1: Relative speed in m/s = distance / time = 125 / 10 = 12.5 m/s. Step 2: Convert man speed to m/s: 4 km/h = 4 * 5/18 = 20/18 = 1.111... m/s. Step 3: Let train speed in m/s be v. Step 4: Relative speed v - 1.111... = 12.5. Step 5: So v = 12.5 + 1.111... ≈ 13.611... m/s. Step 6: Convert v back to km/h: v km/h = 13.611... * 18/5 ≈ 48.9996 km/h. Step 7: Round this to the nearest whole number: approximately 49 km/h. Verification / Alternative check: Take train speed as exactly 49 km/h. Convert to m/s: 49 * 5/18 ≈ 13.611 m/s. Relative speed with respect to the man is then 13.611 - 1.111 ≈ 12.5 m/s. Time to cross a 125 m distance at 12.5 m/s is 125 / 12.5 = 10 seconds, which matches the data perfectly. This confirms that 49 km/h is consistent. Why Other Options Are Wrong: Speeds 50, 51, or 52 km/h would give relative speeds slightly larger than 12.5 m/s and hence smaller crossing times, while 48 km/h would give a slower relative speed and a longer time than 10 seconds. Therefore, they do not match the given time exactly and must be rejected. Common Pitfalls: Typical mistakes include adding the speed of the man instead of subtracting it, forgetting to convert km/h to m/s, or treating the man as stationary. Some learners also incorrectly assume that time depends on both train and man lengths, but here only the train length matters because the man is treated as a point. Final Answer: The speed of the train is 49 kmph.

Question 20

A passenger train is moving at a speed of 120 km/h. If the length of the train is 101 metres, how long (in seconds) will it take to completely cross a railway platform that is 154 metres long?

Accepted Answer

Introduction / Context: This is a standard train and platform problem where the train must cover its own length plus the length of the platform in order to completely pass it. Such questions assess the ability to convert units, combine distances, and apply the basic relation speed = distance / time. Here, both the train length and the platform length are given, and we must compute the total time for the train to clear the platform. Given Data / Assumptions: Speed of the train = 120 km/h. Length of the train = 101 m. Length of the platform = 154 m. The train moves at a uniform speed on straight track. To fully cross the platform, the train must travel a distance equal to train length plus platform length. Concept / Approach: First, the speed must be converted from km/h to m/s to match the distance units in metres. Next, the total distance to be covered while crossing the platform is the sum of the train length and platform length. Then we apply the formula time = distance / speed to find the required time, expressed in seconds. Step-by-Step Solution: Step 1: Convert speed from km/h to m/s. Step 2: 120 km/h = 120 * 5/18 m/s = 600/18 m/s = 33.333... m/s. Step 3: Calculate total distance = train length + platform length = 101 + 154 = 255 m. Step 4: Use time = distance / speed. Step 5: Time = 255 / 33.333... ≈ 7.65 seconds. Step 6: Therefore, the train takes approximately 7.65 seconds to cross the platform completely. Verification / Alternative check: We can verify by approximating the speed as 33.33 m/s. Then in 7.65 seconds, distance covered = 33.33 * 7.65 ≈ 255 metres, which equals the sum of the lengths of the train and the platform. This consistency check confirms that our time is correctly computed. Why Other Options Are Wrong: The values 6.24, 6.58, and 7.325 seconds would correspond to smaller distances at the same speed and hence would not allow the train to travel the full 255 metres. The distractor 8.25 seconds implies the train covers more distance than required, which does not match the exact condition that it should just clear the platform. Common Pitfalls: Students often forget to add the platform length to the train length, treating only the train length as the distance. Another frequent mistake is improper conversion between km/h and m/s, or rounding speed too early, which can cause noticeable errors in time. Carefully handling units and total distance prevents such errors. Final Answer: The train will take approximately 7.65 sec to cross the platform completely.

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