Question 1

A 150 metre long train crosses another 210 metre long train running in the opposite direction in 10.8 seconds. If the shorter train crosses a pole in 12 seconds, what is the speed of the longer train in km/h?

Accepted Answer

Introduction / Context: This problem involves two trains moving in opposite directions, which means their relative speed is the sum of their individual speeds. We are given the time they take to cross each other and additional information about how long the shorter train takes to cross a pole. Using these pieces of data, we can find the speed of the longer train in km/h. Given Data / Assumptions: Length of the shorter train = 150 m. Length of the longer train = 210 m. Time to cross each other when moving in opposite directions = 10.8 s. The shorter train crosses a pole in 12 s. Speeds are constant and motion is along straight tracks. Concept / Approach: When a train crosses a pole, the distance is equal to its own length. We first find the speed of the shorter train by using its length and the time taken to cross the pole. Then we use the combined distance of both trains when they cross each other and the time taken to find their relative speed. The speed of the longer train is then obtained by subtracting the speed of the shorter train from the relative speed. Step-by-Step Solution: Step 1: Speed of the shorter train = 150 m / 12 s = 12.5 m/s. Step 2: Total distance covered when trains cross each other = 150 + 210 = 360 m. Step 3: Relative speed when moving in opposite directions = 360 / 10.8 m/s. Step 4: Compute relative speed: 360 / 10.8 = 33.333... m/s. Step 5: Speed of longer train in m/s = relative speed - speed of shorter train = 33.333... - 12.5 = 20.833... m/s. Step 6: Convert to km/h: 20.833... * 18/5 ≈ 75 km/h. Verification / Alternative check: Take longer train speed as 75 km/h, which is 75 * 5/18 = 20.833... m/s. Sum of speeds in opposite directions = 20.833... + 12.5 = 33.333... m/s. Time to cross a combined distance of 360 m is 360 / 33.333... = 10.8 s, matching the given data exactly, which verifies the result. Why Other Options Are Wrong: Speeds such as 80 km/h or 70 km/h would change the relative speed and thus alter the crossing time away from 10.8 s. Options 54 km/h and 45 km/h are too low and would yield relative speeds that are inconsistent with the required crossing time for the given combined length of 360 m. Common Pitfalls: Learners sometimes forget to add the lengths of both trains when they cross each other. Another common mistake is using the difference instead of the sum of speeds for opposite directions. Errors also occur when converting between m/s and km/h or when misreading which train is shorter or longer. Final Answer: The speed of the longer train is 75 kmph.

Question 2

An express train takes 4 seconds to completely enter a tunnel that is 1 km long. If it is travelling at 108 km/h, how long (in seconds) will it take to pass completely through the tunnel, from the moment the front enters until the rear leaves?

Accepted Answer

Introduction / Context: Questions involving trains and tunnels focus on understanding how far the train must travel for its full length to clear the tunnel. Here, we are told how long the train takes just to enter the tunnel and we know the tunnel length and the train speed. From this we can deduce the train length and then compute the total time to pass through the tunnel. Given Data / Assumptions: Time taken for the train to completely enter the tunnel = 4 s. Length of the tunnel = 1 km = 1000 m. Speed of the train = 108 km/h. The train moves at a uniform speed in a straight line. Total time to fully pass the tunnel is the time from when the front enters until the rear exits. Concept / Approach: When the train is just entering the tunnel, the distance covered in that period is equal to the length of the train itself. Once the front reaches the far end of the tunnel, the train must still move by its own length for the rear to exit. Therefore, for complete passage, the train must travel tunnel length plus its own length. We first find train length from the entry time, then use total distance and speed to find the total time. Step-by-Step Solution: Step 1: Convert speed to m/s: 108 km/h = 108 * 5/18 = 30 m/s. Step 2: Distance covered in 4 s while entering = speed * time = 30 * 4 = 120 m. Step 3: Therefore, length of the train = 120 m. Step 4: For complete passage, distance to be covered = tunnel length + train length = 1000 + 120 = 1120 m. Step 5: Time for full passage = total distance / speed = 1120 / 30 ≈ 37.333... s. Step 6: Rounded to two decimal places, time ≈ 37.33 seconds. Verification / Alternative check: We can check consistency. With a speed of 30 m/s, in 37.333... seconds, the train covers 30 * 37.333... ≈ 1120 m, which exactly equals the sum of train length and tunnel length. This confirms that our calculation is correct and matches the description of complete passage through the tunnel. Why Other Options Are Wrong: A time such as 33 s or 32 s would yield a shorter distance than 1120 m at 30 m/s, so the train would not yet be fully out of the tunnel. On the other hand, 40 s would imply that the train covers more distance than necessary. The option 36.43 s does not correspond to a simple fraction of 1120 / 30 and is inconsistent with the exact computation. Common Pitfalls: Many learners mistakenly use only the tunnel length and forget to add the train length when calculating the distance for full passage. Another error is misinterpreting the 4 seconds as the time to cross the entire tunnel, which would lead to an incorrect train length. Working carefully with distance and time relationships prevents such misunderstandings. Final Answer: The train will take approximately 37.33 sec to pass completely through the tunnel.

Question 3

A train for Fatehpur leaves Agra station every 2 hours 30 minutes. An announcement is made that the previous train left 37 minutes ago and that the next train will arrive at 17:00 hours. At what time was this announcement made?

Accepted Answer

Introduction / Context: This is a time schedule and interval problem frequently seen in aptitude tests. Trains are assumed to leave at fixed regular intervals, and we are given information about when the next train is due along with how long ago the previous train left. From this information, we must work backwards and determine the exact clock time at which the announcement was made. Given Data / Assumptions: Trains for Fatehpur leave Agra station every 2 hours 30 minutes. The next train will come at 17:00 hours. The previous train left 37 minutes before the announcement time. All departures follow the same fixed schedule with no delays. Concept / Approach: If trains leave at regular intervals, then the gap between consecutive departures is constant. We know the time of the next departure and the interval, so we can find the time of the previous departure by subtracting the interval from 17:00 hours. Once we know when the previous train left, we subtract the time elapsed since that departure to get the announcement time. Step-by-Step Solution: Step 1: Departure interval = 2 hours 30 minutes = 150 minutes. Step 2: Next departure time given = 17:00 hours. Step 3: Previous departure time = 17:00 minus 2 hours 30 minutes = 14:30 hours. Step 4: The announcement states that the train left 37 minutes ago. Step 5: Therefore, announcement time = previous departure time + 37 minutes = 14:30 + 0:37 = 15:07 hours. Step 6: Thus, the announcement must have been made at 15:07 hours. Verification / Alternative check: We can verify by checking both sides. From 15:07 to 17:00 is 1 hour 53 minutes. Adding the earlier 37 minutes to reach 2 hours 30 minutes confirms the departure interval of 150 minutes between the previous and the next trains. This consistency ensures that the schedule and announcement information match the calculated time. Why Other Options Are Wrong: Times such as 15:00, 15:05, or 15:20 do not keep the exact 2 hour 30 minute gap between the known 17:00 departure and the earlier train when combined with the stated 37 minute offset. The distractor 14:53 does not align correctly with either the interval or the given 37 minute elapsed time. Common Pitfalls: A common mistake is to subtract 37 minutes from 17:00 instead of adding it to the previous departure time. Some learners also misconvert the 2 hour 30 minute interval, or accidentally treat 2 hours 30 minutes as 2.3 hours. Keeping everything in minutes and working stepwise avoids confusion. Final Answer: The announcement was made at 15:07 hrs.

Question 4

A train of length 200 metres is moving at a speed of 72 km/h. How much time (in seconds) will it take to completely cross another train of length 300 metres that is moving in the same direction at 36 km/h?

Accepted Answer

Introduction / Context: This question deals with two trains moving in the same direction at different speeds. To find the crossing time, we need to consider their relative speed and the total distance that must be covered for one train to pass the other completely. This type of train problem is a classic example of using relative motion in aptitude tests. Given Data / Assumptions: Length of the first train = 200 m. Speed of the first train = 72 km/h. Length of the second train = 300 m. Speed of the second train = 36 km/h. Both trains move in the same direction. Speeds are constant and motion is in a straight line. Concept / Approach: When two objects move in the same direction, the effective speed at which one overtakes the other is the difference in their speeds. For one train to completely cross the other, it must cover a distance equal to the sum of both train lengths. We first compute the relative speed in m/s and then use time = distance / speed to find the crossing time in seconds. Step-by-Step Solution: Step 1: Convert speeds from km/h to m/s. Step 2: 72 km/h = 72 * 5/18 = 20 m/s. Step 3: 36 km/h = 36 * 5/18 = 10 m/s. Step 4: Relative speed when moving in the same direction = 20 - 10 = 10 m/s. Step 5: Total distance to be covered = 200 + 300 = 500 m. Step 6: Time to cross = distance / relative speed = 500 / 10 = 50 seconds. Verification / Alternative check: We can confirm by reversing roles. Regardless of which train is considered as overtaking, the relative speed remains 10 m/s and the total distance is 500 m. Hence the crossing time is always 500 / 10 = 50 seconds. This symmetry confirms the robustness of the calculation. Why Other Options Are Wrong: Times like 58, 60, or 62 seconds correspond to smaller relative speeds or larger distances than given, which do not reflect the actual numerical values. The distractor 55 seconds is close but still does not match the exact computation of 500 / 10 = 50, so it is also incorrect. Common Pitfalls: Learners sometimes add the speeds instead of subtracting them for same direction motion, which would be correct only for opposite directions. Forgetting to add both train lengths or mixing units between km/h and m/s are also frequent mistakes. Careful stepwise calculation avoids these issues. Final Answer: The train will take 50 sec to cross the other train completely.

Question 5

A train 900 metres long is running at a speed of 78 km/h. If it crosses a tunnel completely in 1 minute, what is the length of the tunnel in metres?

Accepted Answer

Introduction / Context: This train problem focuses on calculating the length of a tunnel when a train of known length passes through it in a given time. To clear the tunnel completely, the train must travel a distance equal to its own length plus the length of the tunnel. With the speed of the train and the total time provided, we can determine the tunnel length. Given Data / Assumptions: Length of the train = 900 m. Speed of the train = 78 km/h. Time to cross the tunnel completely = 1 minute = 60 s. Speed is uniform and track is straight. Distance covered during the 60 s equals train length plus tunnel length. Concept / Approach: We convert the train speed from km/h to m/s to match the distance in metres. Then we calculate the total distance travelled in 60 seconds. From this total distance, we subtract the known train length to obtain the tunnel length. The key formula used is distance = speed * time. Step-by-Step Solution: Step 1: Convert speed from km/h to m/s. Step 2: 78 km/h = 78 * 5/18 = 390/18 = 21.666... m/s. Step 3: Total distance covered in 60 seconds = 21.666... * 60 ≈ 1300 m. Step 4: This total distance equals train length + tunnel length. Step 5: Tunnel length = 1300 - 900 = 400 m. Step 6: Hence, the tunnel is 400 metres long. Verification / Alternative check: We can recalculate more exactly using fractions. 78 km/h = 78 * 5/18 = 390/18 = 65/3 m/s. In 60 s, distance = (65/3) * 60 = 65 * 20 = 1300 m. Subtracting the train length 900 m gives 400 m exactly. The arithmetic confirms the tunnel length without relying on decimal approximations. Why Other Options Are Wrong: A tunnel of 300 m or 350 m would result in a total distance of 1200 or 1250 m respectively, which does not match the calculated 1300 m. Options 500 m and 600 m give total distances of 1400 and 1500 m, which are too large for the same speed and time combination. Common Pitfalls: Learners sometimes mistakenly use only the tunnel length as the distance for complete crossing, forgetting that the train must also move its full length. Another pitfall is incorrect conversion from km/h to m/s or using 60 minutes instead of 60 seconds. Keeping track of units throughout the solution is crucial. Final Answer: The length of the tunnel is 400 m.

Question 6

A train covers k kilometres at 40 km/h and then another 2k kilometres at 20 km/h. What is the average speed of the train in km/h for the entire 3k kilometres of the journey?

Accepted Answer

Introduction / Context: Average speed questions often appear in aptitude tests to check whether candidates understand that average speed is total distance divided by total time, not the simple average of two speeds. Here, a train travels different parts of its journey at different speeds. We must carefully calculate the total time taken and then use the average speed formula over the entire distance of 3k kilometres. Given Data / Assumptions: First part of the journey: distance = k km, speed = 40 km/h. Second part of the journey: distance = 2k km, speed = 20 km/h. Total distance travelled = k + 2k = 3k km. Speeds are constant in each segment. Concept / Approach: Average speed is defined as total distance divided by total time. We first compute the time taken for each part of the trip using time = distance / speed. After adding these times, we divide the total distance, which is 3k km, by this total time. The variable k will cancel out, giving a numeric value for the average speed. Step-by-Step Solution: Step 1: Time for first part = distance / speed = k / 40 hours. Step 2: Time for second part = 2k / 20 hours. Step 3: Simplify second time: 2k / 20 = k / 10 hours. Step 4: Total time = k / 40 + k / 10. Step 5: Express with common denominator: k / 40 + 4k / 40 = 5k / 40 = k / 8 hours. Step 6: Total distance = 3k km. Step 7: Average speed = total distance / total time = 3k / (k / 8) = 3k * 8 / k = 24 km/h. Verification / Alternative check: We can choose a specific convenient value, for example k = 40 km. Then first leg: 40 km at 40 km/h takes 1 hour. Second leg: 80 km at 20 km/h takes 4 hours. Total distance = 120 km, total time = 5 hours. Average speed = 120 / 5 = 24 km/h, which matches the derived result and confirms that k cancels correctly. Why Other Options Are Wrong: A simple average of 40 and 20 is 30 km/h, which is incorrect because the train spends more time at the lower speed of 20 km/h. Values such as 26, 28, or 32 km/h do not satisfy the total distance and total time relationship when recalculated carefully, so they must be rejected. Common Pitfalls: The most common mistake is to take the arithmetic mean of 40 and 20 instead of using total distance divided by total time. Another error is mishandling algebra with k or failing to harmonise denominators when adding times. Always remember that average speed depends on how long the train travels at each speed, not just the numerical values of the speeds themselves. Final Answer: The average speed of the train for the entire journey is 24 kmph.

Question 7

Two trains start simultaneously, one from Hyderabad to Bangalore and the other from Bangalore to Hyderabad. After they meet, the trains reach their respective destinations in 9 hours and 16 hours respectively. What is the ratio of the speed of the f…

Accepted Answer

Introduction / Context: This is a classic meeting trains question where two trains start from opposite ends of a track and meet somewhere in between. We are not given actual distances or speeds, but we are told how long each train takes to reach its destination after they meet. Using this information, we can deduce the ratio of their speeds without knowing the actual length of the route between Hyderabad and Bangalore. Given Data / Assumptions: Train 1 starts from Hyderabad towards Bangalore. Train 2 starts from Bangalore towards Hyderabad. They start at the same time and meet at some point on the route. After meeting, Train 1 takes 9 hours to reach its destination. After meeting, Train 2 takes 16 hours to reach its destination. Speeds are constant and the track is straight and fixed in length. Concept / Approach: Let the speeds of the two trains be v1 and v2 respectively, and let the time taken until they meet be T hours. At the meeting point, the distance remaining for Train 1 to reach its destination equals the distance already covered by Train 2, and vice versa. A known result from such problems is that the speeds of the trains are inversely proportional to the times they take to reach their destinations after meeting, more precisely v1 / v2 = square root of (t2 / t1). We derive and apply this relationship. Step-by-Step Solution: Step 1: Let time from start to meeting be T hours. Step 2: Distance covered by Train 1 before meeting = v1 * T. Step 3: Distance covered by Train 2 before meeting = v2 * T. Step 4: After meeting, Train 1 still needs to cover distance v2 * T and takes 9 hours to cover it, so v2 * T = v1 * 9. Step 5: Similarly, Train 2 must cover distance v1 * T and takes 16 hours, so v1 * T = v2 * 16. Step 6: Divide these two equations: (v2 * T) / (v1 * T) = (v1 * 9) / (v2 * 16). Step 7: Simplify: v2 / v1 = (v1 * 9) / (v2 * 16). Step 8: Cross multiply: (v2)^2 * 16 = (v1)^2 * 9. Step 9: Therefore, (v1 / v2)^2 = 16 / 9, so v1 / v2 = 4 / 3. Verification / Alternative check: We can take v1 = 4 units and v2 = 3 units. From equation v2 * T = v1 * 9, we get 3T = 4 * 9 = 36, so T = 12 hours. Then v1 * T = 4 * 12 = 48 units. From the other equation v1 * T = v2 * 16, we get 48 = 3 * 16, which is also 48, confirming consistency and validating the ratio 4 : 3. Why Other Options Are Wrong: Ratios like 3 : 4 or 2 : 3 do not satisfy the set of equations that link distances and times after meeting. The option 3 : 2 would correspond to the square root of 9 : 4, not 16 : 9, and 5 : 4 has no algebraic support here. Only 4 : 3 matches the derived relationship between speeds and times after meeting. Common Pitfalls: Learners often assume speeds are simply inversely proportional to times after meeting (3 : 4) without considering the squared relationship that emerges from the symmetry of distances. Others try to apply distance formulas incorrectly or forget that both trains travel the same total route. A careful derivation, as above, helps avoid these errors. Final Answer: The ratio of the speeds of the two trains is 4 : 3.

Question 8

The length of a bridge which a train 130 metres long, travelling at 45 km/h, can completely cross in 30 seconds is equal to how many metres?

Accepted Answer

Introduction / Context: This question is about a train crossing a bridge, a classic type of quantitative aptitude problem. For the train to completely cross the bridge, the total distance it travels from the moment its front reaches the start of the bridge until its rear leaves the far end equals the sum of the train length and the bridge length. We know the train speed, the train length, and the time taken, so we can compute the bridge length. Given Data / Assumptions: Length of the train = 130 m. Speed of the train = 45 km/h. Time taken to completely cross the bridge = 30 s. Speed is uniform and the track is straight. Total distance covered in 30 s = train length + bridge length. Concept / Approach: We first convert the speed from km/h to m/s to work consistently with metres and seconds. Then we use the formula distance = speed * time to find how far the train moves in 30 seconds. The bridge length is the excess of this total distance over the known train length. This method directly applies the idea that the train must clear its own length plus the bridge length. Step-by-Step Solution: Step 1: Convert 45 km/h to m/s using 1 km/h = 5/18 m/s. Step 2: 45 km/h = 45 * 5/18 = 225/18 = 12.5 m/s. Step 3: Total distance covered in 30 s = speed * time = 12.5 * 30 = 375 m. Step 4: This total distance equals train length + bridge length. Step 5: Bridge length = total distance - train length = 375 - 130 = 245 m. Step 6: Therefore, the bridge is 245 metres long. Verification / Alternative check: Check by reversing the calculation. If the bridge is 245 m and the train is 130 m, then total distance is 375 m. At 12.5 m/s, time needed = 375 / 12.5 = 30 s, which matches the given time. This confirms that our bridge length is consistent with the problem statement. Why Other Options Are Wrong: Bridge lengths such as 196, 210, 220, or 230 m would all lead to total distances less than 375 m, which, at 12.5 m/s, would result in times less than 30 seconds. Since the question explicitly states a crossing time of 30 seconds, these values cannot be correct. Common Pitfalls: Learners sometimes forget to add the train length when computing the distance needed to cross a bridge, using only the bridge length instead. Others may incorrectly convert 45 km/h to m/s or miscalculate the product of speed and time. Following the formula carefully with correct unit conversions avoids these errors. Final Answer: The length of the bridge is 245 m.

Question 9

Train X crosses a stationary train Y in 60 seconds and crosses a pole in 25 seconds at the same constant speed. If the length of train X is 300 metres, what is the length of stationary train Y in metres?

Accepted Answer

Introduction / Context: Here we compare how a moving train crosses a stationary train and a fixed pole. Crossing a pole reveals the speed of train X because only its own length matters. Crossing a stationary train requires train X to cover the combined length of both trains. By using the same speed and the different times, we can determine the length of train Y. Given Data / Assumptions: Train X length = 300 m. Train X crosses a pole in 25 s. Train X crosses stationary train Y in 60 s. Train Y is stationary. Speed of train X is constant throughout. Concept / Approach: When a train crosses a pole, distance = length of the train, so we can find its speed from length and time. When it crosses another stationary train, the required distance is length of X plus length of Y. Time and speed are known, so we can solve for the length of Y. The key formulas are speed = distance / time and distance = speed * time. Step-by-Step Solution: Step 1: Speed of train X when crossing a pole = length / time = 300 / 25 = 12 m/s. Step 2: When crossing train Y, distance to be covered = length of X + length of Y. Step 3: Let length of Y be L metres. Step 4: Time to cross Y = 60 s, speed = 12 m/s. Step 5: Therefore, distance = speed * time = 12 * 60 = 720 m. Step 6: So 300 + L = 720. Step 7: Hence, L = 720 - 300 = 420 m. Verification / Alternative check: We can verify by using the derived lengths. If train X is 300 m and train Y is 420 m, total distance is 720 m. At 12 m/s, time required is 720 / 12 = 60 s, which matches the given crossing time. For the pole, 300 m at 12 m/s gives 25 s, confirming consistency with the original data. Why Other Options Are Wrong: Lengths like 360, 380, or 320 m would lead to total distances differing from 720 m, and thus crossing times different from 60 s at 12 m/s. The value 460 m also fails this check, as 300 + 460 = 760 m, which would require more than 60 seconds to cross at the same speed. Common Pitfalls: A frequent error is to consider only one of the train lengths when crossing another train instead of summing both. Some learners may miscalculate the speed of train X or misapply the distance = speed * time relation. Keeping a clear distinction between pole crossing and train crossing scenarios is vital. Final Answer: The length of stationary train Y is 420 m.

Question 10

A train travels 360 kilometres at a uniform speed. If its speed had been 5 km/h more, it would have taken 1 hour less to cover the same distance. What is the actual speed of the train in km/h?

Accepted Answer

Introduction / Context: This is a classic average speed and time relationship problem where changing the speed changes the time taken. The distance is fixed, and we compare the actual travel time to a hypothetical travel time at a slightly higher speed. Such questions test algebraic manipulation of the formula time = distance / speed. Given Data / Assumptions: Actual distance travelled = 360 km. Actual speed of the train = v km/h (unknown). Hypothetical speed = v + 5 km/h. At speed v + 5 km/h, the time taken is 1 hour less than at speed v. Speeds are uniform and no stops are considered. Concept / Approach: We express the original time in terms of v using time = distance / speed. Similarly, we express the new time with speed v + 5. The condition that the new time is 1 hour less than the original provides an equation relating v. Solving this equation yields the actual speed. Finally, we verify by checking both travel times. Step-by-Step Solution: Step 1: Let original speed be v km/h. Step 2: Original time = 360 / v hours. Step 3: New speed = v + 5 km/h, new time = 360 / (v + 5) hours. Step 4: Given condition: 360 / v - 360 / (v + 5) = 1. Step 5: Multiply through by v(v + 5) to clear denominators: 360(v + 5) - 360v = v(v + 5). Step 6: Simplify left side: 360v + 1800 - 360v = 1800. Step 7: So equation becomes 1800 = v^2 + 5v. Step 8: Rearrange: v^2 + 5v - 1800 = 0. Step 9: Factorise: (v - 40)(v + 45) = 0, so v = 40 or v = -45. Step 10: Negative speed is not meaningful, so v = 40 km/h. Verification / Alternative check: At 40 km/h, time = 360 / 40 = 9 hours. At 45 km/h, time = 360 / 45 = 8 hours. The difference in time is 9 - 8 = 1 hour, which matches the problem statement exactly. Therefore, 40 km/h is confirmed as the correct actual speed. Why Other Options Are Wrong: Speeds such as 42, 43, or 45 km/h do not satisfy the exact condition of 1 hour difference when recalculated. For example, 42 km/h and 47 km/h or 45 km/h and 50 km/h will not give the required difference with the fixed distance of 360 km. The distractor 35 km/h also fails the algebraic equation derived above. Common Pitfalls: Learners may attempt to guess speeds instead of forming and solving the equation, which can be time consuming and inaccurate. Another mistake is to incorrectly handle algebraic manipulation or to forget that time difference is given as 1 hour, leading to misinterpretation. Writing out the equation step by step and solving carefully prevents such errors. Final Answer: The actual speed of the train is 40 kmph.

Question 11

Two trains are moving in opposite directions at speeds of 30 km/h and 45 km/h respectively. Their lengths are 450 metres and 550 metres respectively. How many seconds does the slower train take to completely cross the faster train?

Accepted Answer

Introduction / Context: This question involves two trains moving in opposite directions, which means their relative speed is the sum of their individual speeds. To find how long one train takes to cross the other, we must consider the total distance (sum of their lengths) and the relative speed. The result will be the time during which they completely pass each other. Given Data / Assumptions: Speed of the slower train = 30 km/h. Speed of the faster train = 45 km/h. Length of the slower train = 450 m. Length of the faster train = 550 m. The trains move in opposite directions. Speeds are constant and track is straight. Concept / Approach: When two bodies move in opposite directions, their relative speed equals the sum of their individual speeds. For one train to fully cross the other, it must cover a distance equal to the sum of both lengths. We convert speeds from km/h to m/s, add them to get relative speed, and then use time = distance / speed to find the crossing time in seconds. Step-by-Step Solution: Step 1: Total distance to be covered = 450 + 550 = 1000 m. Step 2: Convert speeds to m/s using 1 km/h = 5/18 m/s. Step 3: 30 km/h = 30 * 5/18 = 150/18 = 8.333... m/s. Step 4: 45 km/h = 45 * 5/18 = 225/18 = 12.5 m/s. Step 5: Relative speed = 8.333... + 12.5 = 20.833... m/s. Step 6: Time to cross = distance / relative speed = 1000 / 20.833... seconds. Step 7: Calculate: 1000 / (75 * 5/18) can be simplified as 1000 / (75 * 1000 / 3600) = 3600 / 150 = 48 seconds. Verification / Alternative check: Using the exact relation, relative speed = (30 + 45) km/h = 75 km/h. Time in hours = distance in km / speed in km/h = 1 km / 75 km/h = 1/75 hours. Converting to seconds: (1/75) * 3600 = 3600 / 75 = 48 seconds, which matches the earlier result and confirms correctness. Why Other Options Are Wrong: Values such as 54, 62, or 72 seconds would correspond to smaller relative speeds than 75 km/h for the same 1 km distance. The distractor 52 seconds is close but cannot be obtained from the exact speed and distance relationship. Only 48 seconds satisfies the mathematical relationship precisely. Common Pitfalls: Some learners forget to convert metres to kilometres or mix units, which leads to incorrect answers. Others may mistakenly subtract the train speeds even though the trains move in opposite directions. Remember to add speeds for opposite directions and to include the sum of lengths when calculating the distance for crossing. Final Answer: The slower train takes 48 seconds to completely cross the faster train.

Question 12

A train 160 metres long is running at 72 km/h. How much time (in seconds) will it take to cross an electric pole completely?

Accepted Answer

Introduction / Context: This is a basic train and pole question, serving as a foundation for more complex problems. When a train crosses a pole, it needs to cover only its own length. With the train length and its speed given, we can directly use the distance and speed relationship to find the time taken to cross the pole completely. Given Data / Assumptions: Length of the train = 160 m. Speed of the train = 72 km/h. The pole is treated as a point object. Speed is uniform and track is straight. Concept / Approach: For a train crossing a pole, distance = length of the train. We convert the speed from km/h to m/s, ensuring that units match. Then we apply the formula time = distance / speed to get the answer in seconds. This is a direct one step calculation once units are consistent. Step-by-Step Solution: Step 1: Convert 72 km/h to m/s. Step 2: 72 km/h = 72 * 5/18 = 20 m/s. Step 3: Distance to be covered to cross the pole = train length = 160 m. Step 4: Use time = distance / speed. Step 5: Time = 160 / 20 = 8 seconds. Verification / Alternative check: We can think in reverse. If the train takes 8 seconds at 20 m/s, the distance covered is 20 * 8 = 160 m, which exactly equals the given train length. This perfectly matches the definition of crossing a pole, confirming that 8 seconds is correct. Why Other Options Are Wrong: A time of 4 or 6 seconds would imply higher speeds than 72 km/h for the same train length, while 9 or 10 seconds would imply lower speeds. Since the speed is fixed at 72 km/h, these alternatives are inconsistent with the distance and speed relationship. Common Pitfalls: Learners sometimes forget to convert km/h to m/s or accidentally use 72 / 20 instead of 160 / 20 when computing time. Additionally, some confuse the distance for a pole with that for a platform or another train, where more than one length is involved. Being clear about what the train is crossing avoids such confusion. Final Answer: The train will take 8 sec to cross the electric pole completely.

Question 13

Excluding stoppages, the speed of a train is 120 km/h, but including stoppages its effective speed becomes 50 km/h. For how many minutes in each hour does the train stop?

Accepted Answer

Introduction / Context: This problem explores the concept of effective speed when a train makes stoppages. The train moves at a higher speed while in motion, but due to stops its overall average speed reduces. By comparing the distance it could have travelled without stoppages to the actual distance covered including stops, we can determine the amount of time it spends stopped in each hour. Given Data / Assumptions: Speed when the train is moving (excluding stops) = 120 km/h. Effective average speed including stops = 50 km/h. We seek the total stopping time per hour of clock time. Speed while moving is constant. Concept / Approach: Assume a reference period of 1 hour of clock time. During this hour, the train moves for some time and stands still for the rest. When moving, it covers distance at 120 km/h, but because of stoppages the overall distance in one hour is only 50 km. Let the running time be t hours and the stopping time be 1 - t hours. We use the relationship distance = speed * time with the effective speed condition to solve for t and then compute the stopping time in minutes. Step-by-Step Solution: Step 1: Let total clock period considered = 1 hour. Step 2: Let the train run for t hours and stop for (1 - t) hours in that hour. Step 3: Distance covered while running = 120 * t km. Step 4: Effective distance in 1 hour = 50 km (since effective speed is 50 km/h). Step 5: Therefore, 120 * t = 50. Step 6: Solve for t: t = 50 / 120 = 5/12 hours. Step 7: Stopping time = 1 - 5/12 = 7/12 hours. Step 8: Convert stopping time to minutes: (7/12) * 60 = 7 * 5 = 35 minutes. Verification / Alternative check: If the train runs for 5/12 hours (25 minutes) at 120 km/h, the distance covered is 120 * 5/12 = 50 km. In the remaining 7/12 hours (35 minutes) it is stationary and covers no distance, so the total distance in 1 hour is still 50 km, giving an effective speed of 50 km/h, which matches the problem statement. Why Other Options Are Wrong: Stopping times of 20, 25, 30, or 40 minutes would all give different effective speeds when recalculated. For example, stopping 20 minutes implies running for 40 minutes at 120 km/h, which would yield 80 km in an hour, not 50 km. Only a 35 minute stoppage per hour fits all the given data. Common Pitfalls: Some learners misinterpret the speeds and attempt to average them directly, while others forget that effective speed is based on total distance over total clock time. Confusing the fraction of time stopped with the fraction of distance lost also leads to errors. Treating 1 hour as a reference and carefully applying the distance formula avoids these problems. Final Answer: The train stops for 35 minutes in each hour.

Question 14

A train moving at two thirds of its normal speed reaches its destination 20 minutes late. What is the normal time, in hours, that the train usually takes to cover this journey?

Accepted Answer

Introduction / Context: This question examines the relationship between speed and time for a fixed distance, a common theme in quantitative reasoning. If speed is reduced, time increases proportionally. We are told that the train runs at two thirds of its normal speed and therefore arrives 20 minutes late. Using proportionality, we can compute the normal travel time for the journey. Given Data / Assumptions: Normal speed of the train = v (some value in km/h). Normal time for the journey = T hours (unknown). Reduced speed = (2/3) * v. At the reduced speed, the train arrives 20 minutes late. Distance of the journey is fixed. Concept / Approach: For a fixed distance, speed and time are inversely proportional: speed * time = constant. If speed is multiplied by some factor, time is divided by the same factor. Here, speed becomes two thirds of normal, so time becomes three halves of normal. The difference between the new time and the original time is given as 20 minutes. Setting up an equation in T using this relationship allows us to solve for the normal time. Step-by-Step Solution: Step 1: Let normal time be T hours. Step 2: Normal speed is v, so distance = v * T. Step 3: Reduced speed = (2/3) * v. Step 4: New time with reduced speed = distance / reduced speed = (v * T) / [(2/3) * v] = T * (3/2) hours. Step 5: Increase in time = new time - normal time = (3/2) * T - T = (1/2) * T. Step 6: This increase is given as 20 minutes = 20/60 = 1/3 hour. Step 7: So (1/2) * T = 1/3. Step 8: Multiply both sides by 2: T = 2/3 hours. Verification / Alternative check: If normal time is 2/3 hours (which is 40 minutes), then new time at reduced speed is (3/2) * 2/3 = 1 hour = 60 minutes. The delay is 60 - 40 = 20 minutes, which matches the given condition. This confirms that the computed normal time is consistent with the situation described. Why Other Options Are Wrong: Values such as 4/3, 3/2, or 1/4 hours would produce different time differences when multiplied by 3/2 and then subtracted from the original. None of them lead to an exact 20 minute difference except 2/3 hours. The distractor 1 hour clearly gives no delay when scaled, contradicting the problem statement. Common Pitfalls: A typical mistake is to misinterpret the factor relating speed and time, for example assuming time changes by two thirds instead of three halves. Another error is failing to convert 20 minutes into hours correctly, which leads to incorrect equations. Carefully using the idea that speed * time is constant for a fixed distance avoids these errors. Final Answer: The normal time for the journey is 2/3 hrs (that is, 40 minutes).

Question 15

A jogger running at 9 km/h along a railway track is 260 metres ahead of the engine of a 140 metre long train that is running at 45 km/h in the same direction. In how many seconds will the train completely pass the jogger?

Accepted Answer

Introduction / Context: This problem involves relative speed between a train and a jogger moving in the same direction along a railway track. The jogger is initially ahead of the train by a certain distance. For the train to completely pass the jogger, the engine must cover the gap plus the full length of the train. Using relative speed and distance ideas, we can find the time taken. Given Data / Assumptions: Jogger speed = 9 km/h. Train speed = 45 km/h. Jogger is 260 m ahead of the engine initially. Train length = 140 m. Both move in the same direction at constant speeds. Concept / Approach: When two objects move in the same direction, their relative speed is the difference of their speeds. For the train to completely pass the jogger, the engine must first close the 260 metre gap and then the entire 140 metre length of the train must pass in front of the jogger. Hence, the total distance that must be covered relative to the jogger is 260 + 140 metres. We compute the relative speed in m/s and then apply time = distance / speed. Step-by-Step Solution: Step 1: Total relative distance to be covered = 260 + 140 = 400 m. Step 2: Convert speeds from km/h to m/s. Step 3: Train speed = 45 km/h = 45 * 5/18 = 12.5 m/s. Step 4: Jogger speed = 9 km/h = 9 * 5/18 = 2.5 m/s. Step 5: Relative speed = 12.5 - 2.5 = 10 m/s. Step 6: Time taken to pass = distance / relative speed = 400 / 10 = 40 seconds. Verification / Alternative check: In 40 seconds, the relative distance covered at 10 m/s is exactly 400 m, equal to the initial head start plus the train length. At that moment, the rear end of the train has just passed the jogger. Any smaller or larger time would either not fully clear the jogger or overshoot, which confirms that 40 seconds is the correct exact answer. Why Other Options Are Wrong: Times like 25, 32, or 38 seconds correspond to smaller distances than 400 m at 10 m/s, so the train would not have fully passed the jogger. The distractor 30 seconds gives only 300 m of relative displacement, again insufficient. Therefore, only 40 seconds satisfies the exact distance requirement. Common Pitfalls: Learners may forget to add the train length to the initial separation or may incorrectly add the two speeds instead of subtracting them for same direction motion. Mixing up metres and kilometres or failing to convert speeds properly can also lead to errors. Keeping careful track of relative speed and total distance to be covered solves the problem correctly. Final Answer: The train will completely pass the jogger in 40 sec.

Question 16

Two trains are moving in opposite directions at 60 km/h and 90 km/h respectively. Their lengths are 1.10 km and 0.90 km. What time, in seconds, does the slower train take to completely cross the faster train?

Accepted Answer

Introduction / Context: This question extends standard train crossing problems by giving lengths in kilometres instead of metres. Two trains move in opposite directions, so their relative speed is the sum of their speeds. To find the crossing time, we consider the total distance that must be covered (sum of their lengths) and divide it by their relative speed, taking care to keep units consistent. Given Data / Assumptions: Speed of the slower train = 60 km/h. Speed of the faster train = 90 km/h. Length of the slower train = 1.10 km. Length of the faster train = 0.90 km. The trains move in opposite directions at constant speeds. Concept / Approach: When two objects move towards each other on a straight line, their relative speed equals the sum of their speeds. The total distance that needs to be covered for them to completely cross each other is the sum of their lengths. Here, all data can be kept in kilometres and hours initially, and the final time can then be converted to seconds. The core formula is time = distance / speed. Step-by-Step Solution: Step 1: Total distance to be covered when they cross each other = 1.10 + 0.90 = 2.00 km. Step 2: Relative speed in opposite directions = 60 + 90 = 150 km/h. Step 3: Time in hours = distance / speed = 2.00 / 150 hours. Step 4: Simplify: 2 / 150 = 1 / 75 hours. Step 5: Convert time into seconds: (1 / 75) * 3600 seconds. Step 6: Compute: 3600 / 75 = 48 seconds. Verification / Alternative check: We can verify by converting speeds to m/s and lengths to metres. Total distance = 2.00 km = 2000 m. Relative speed = 150 km/h = 150 * 5/18 = 41.666... m/s. Time = 2000 / 41.666... ≈ 48 seconds. This agrees exactly with the previous calculation, confirming the result. Why Other Options Are Wrong: Times like 42, 44, 46, or 50 seconds correspond to different effective relative speeds when recalculated against the fixed distance of 2.00 km. None of them match the given speeds of 60 km/h and 90 km/h when checked rigorously. Only 48 seconds fits the precise relationship between distance and relative speed. Common Pitfalls: Common errors include forgetting to convert kilometres to metres or miscalculating the relative speed by subtracting instead of adding for opposite directions. Some learners may also misuse unit conversion between hours and seconds. Carefully keeping track of units and remembering that relative speed is the sum for opposite directions prevents these mistakes. Final Answer: The slower train takes 48 sec to completely cross the faster train.

Question 17

A 260 m long train is running at 120 km/h and crosses another train coming from the opposite direction at 80 km/h in 9 seconds. What is the length (in metres) of the second train?

Accepted Answer

Introduction / Context: This is a classic aptitude question on trains and relative speed. The goal is to use the concept of relative speed in opposite directions to find the unknown length of the second train, given the crossing time and the speed of both trains. Given Data / Assumptions: Length of first train = 260 m. Speed of first train = 120 km/h. Speed of second train = 80 km/h. They move in opposite directions. Time taken to completely cross each other = 9 seconds. We assume both trains move at constant speeds in a straight line. Concept / Approach: When two objects move towards each other on a straight track, their relative speed is the sum of their individual speeds. The total distance covered during the crossing is the sum of their lengths. Using speed = distance / time, we first compute the relative speed in metres per second, then multiply by the time to get the total distance, and finally subtract the known length of the first train to obtain the length of the second train. Step-by-Step Solution: Step 1: Relative speed in km/h = 120 + 80 = 200 km/h. Step 2: Convert 200 km/h to m/s using 1 km/h = 5/18 m/s. Step 3: Relative speed = 200 * 5 / 18 = 1000 / 18 m/s. Step 4: Total distance covered during crossing = relative speed * time. Step 5: Distance = (1000 / 18) * 9 = 500 m. Step 6: Sum of lengths of both trains = 500 m. Step 7: Length of second train = 500 - 260 = 240 m. Verification / Alternative check: We can verify by reversing the process. If the second train is 240 m, total length becomes 260 + 240 = 500 m. With a relative speed of 1000 / 18 m/s, the time is distance / speed = 500 / (1000 / 18) = 500 * 18 / 1000 = 9 seconds, which matches the given condition. So the answer is consistent. Why Other Options Are Wrong: 250 m: This gives total length 510 m, which would require slightly more than 9 seconds, so it does not fit. 260 m: This gives total length 520 m and would also increase the time above 9 seconds. 270 m: This gives total length 530 m, making the required time even larger than 9 seconds. Common Pitfalls: Students often subtract speeds instead of adding them for opposite directions. Another common mistake is to forget to convert km/h into m/s, which leads to incorrect distances. Sometimes learners wrongly assume that only one train length is involved when two trains are crossing, but in such problems the distance covered during crossing is always the sum of both lengths. Final Answer: Therefore, the length of the second train is 240 m.

Question 18

Two trains start from stations M and N towards each other with speeds of 54 km/h and 63 km/h respectively, and when they meet the second train has travelled 100 km more than the first. What is the distance (in km) between M and N?

Accepted Answer

Introduction / Context: This aptitude question tests understanding of relative motion and simultaneous equations in the context of two trains travelling towards each other. We are given their speeds and the difference in distances travelled when they meet, and we must determine the total distance between the two stations. Given Data / Assumptions: Speed of train from M = 54 km/h. Speed of train from N = 63 km/h. They move towards each other. When they meet, distance covered by second train is 100 km more than that of the first train. Both trains start at the same time and move at constant speeds. Concept / Approach: Let the time taken for the trains to meet be t hours. Distance covered by each train is speed multiplied by time. By using the information about their distance difference, we can find t. Once we know t, we calculate the distance between the stations as the sum of the distances covered by both trains until meeting. Step-by-Step Solution: Step 1: Let time to meet be t hours. Step 2: Distance by first train = 54 * t km. Step 3: Distance by second train = 63 * t km. Step 4: Given that second train travels 100 km more, so 63 * t = 54 * t + 100. Step 5: Simplify: 63t - 54t = 100 which gives 9t = 100. Step 6: Therefore t = 100 / 9 hours. Step 7: Distance between stations = 54t + 63t = (54 + 63) * t = 117 * (100 / 9). Step 8: Compute 117 * 100 / 9 = 13 * 100 = 1300 km. Verification / Alternative check: Check each individual distance. For t = 100 / 9 hours, distance by first train is 54 * 100 / 9 = 600 km. Distance by second train is 63 * 100 / 9 = 700 km. Their sum is 1300 km, and the difference 700 - 600 = 100 km matches the given condition. Hence the calculation is correct. Why Other Options Are Wrong: 600 km: This would make each train cover much smaller distances and the difference could not be 100 km as stated. 700 km: This distance is less than the distance covered by the two trains together in the correct solution. 1500 km: This would require a different time or speed relationship and does not satisfy the distance difference of 100 km. Common Pitfalls: A common mistake is to confuse the difference in speeds with the difference in distances. Another frequent error is to try to use relative speed directly without setting up the equation for the given extra 100 km. Forgetting that both trains travel for the same time until they meet is another typical issue in such problems. Final Answer: Thus, the distance between stations M and N is 1300 km.

Question 19

A train travelling at 90 km/h takes 10 seconds to cross a bridge of length 100 m. What is the length (in metres) of the train?

Accepted Answer

Introduction / Context: This quantitative aptitude question is about finding the length of a train using its speed and the time taken to cross a bridge. It combines basic speed, distance, and time concepts that are very common in problems on trains. Given Data / Assumptions: Speed of the train = 90 km/h. Length of the bridge = 100 m. Time taken to completely cross the bridge = 10 seconds. The train moves at constant speed in a straight line. The train length plus bridge length is the total distance covered during the crossing. Concept / Approach: To find the train length, we calculate how much distance the train covers in 10 seconds at a speed of 90 km/h. The train must cover its own length plus the length of the bridge, so total distance equals train length plus 100 m. We first convert the speed from km/h to m/s, then use distance = speed * time, and finally subtract the bridge length to get the train length. Step-by-Step Solution: Step 1: Convert 90 km/h to m/s using the factor 5/18. Step 2: Speed in m/s = 90 * 5 / 18 = 25 m/s. Step 3: Distance covered in 10 seconds = speed * time = 25 * 10 = 250 m. Step 4: Total distance while crossing the bridge = train length + bridge length. Step 5: So train length + 100 = 250. Step 6: Train length = 250 - 100 = 150 m. Verification / Alternative check: Recheck by forward calculation. If the train is 150 m long and the bridge is 100 m, total distance is 250 m. At 25 m/s, time = 250 / 25 = 10 seconds, which matches the given time exactly. Thus the train length is correctly found. Why Other Options Are Wrong: 130 m: This would give total distance 230 m and a time less than 10 seconds. 120 m: This would give only 220 m, leading to a crossing time of 220 / 25 = 8.8 seconds. 140 m: This leads to 240 m, giving 240 / 25 = 9.6 seconds, still not equal to 10 seconds. Common Pitfalls: Many learners forget that the train must travel its own length plus the bridge length when crossing. Another common error is failing to convert km/h to m/s correctly, which leads to incorrect numbers. It is also important not to mix units and to always keep distances in metres and time in seconds when using m/s. Final Answer: The length of the train is 150 m.

Question 20

A goods train passes a station, and 5 hours later another train travelling at 70 km/h in the same direction passes the same station and then overtakes the goods train in 9 hours. What is the speed (in km/h) of the goods train?

Accepted Answer

Introduction / Context: This train problem focuses on relative speed in the same direction and the idea of a head start. A goods train leaves first, and a faster train follows later, eventually overtaking it. The question asks for the speed of the slower, goods train based on the given timings. Given Data / Assumptions: The goods train passes the station first. Second train speed = 70 km/h. The second train passes the station 5 hours after the goods train. From the moment it passes the station, the faster train takes 9 hours to overtake the goods train. Both trains move in the same direction at constant speeds. Concept / Approach: Let the speed of the goods train be v km/h. When the faster train starts from the station, the goods train already has a time lead and is ahead by a certain distance. The faster train catches up because its speed is higher. The head start distance equals speed of goods train multiplied by 5 hours. The closing speed is 70 - v. Using time = distance / relative speed, we form an equation and solve for v. Step-by-Step Solution: Step 1: Let speed of goods train be v km/h. Step 2: In 5 hours, goods train travels a head start distance = 5 * v km. Step 3: From the moment the second train passes the station, it takes 9 hours to catch the goods train. Step 4: Relative speed (closing speed) = 70 - v km/h. Step 5: Distance to be closed = head start distance = 5v. Step 6: Use formula: distance = relative speed * time, so 5v = (70 - v) * 9. Step 7: Simplify: 5v = 630 - 9v, so 14v = 630 and v = 630 / 14 = 45 km/h. Verification / Alternative check: At 45 km/h, in 5 hours the goods train goes 225 km ahead. Relative speed of the faster train is 70 - 45 = 25 km/h. Time to cover 225 km at 25 km/h is 225 / 25 = 9 hours, which agrees with the given data. Thus the value is correct. Why Other Options Are Wrong: 54 km/h: This would give head start 270 km and relative speed 16 km/h, requiring 270 / 16 hours, which is not 9. 67.5 km/h: This is too close to 70 km/h, giving very small relative speed and far more than 9 hours. 36 km/h: This gives head start 180 km and relative speed 34 km/h, and 180 / 34 is not equal to 9. Common Pitfalls: Learners sometimes forget that the goods train has moved for 5 extra hours before the second train starts. Another mistake is incorrectly using 70 + v instead of 70 - v for the relative speed because the trains move in the same direction. It is also easy to mix up which time interval corresponds to which portion of motion, so handling time references carefully is important. Final Answer: The speed of the goods train is 45 km/h.

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