Question 1

A man standing on a platform finds that a train takes 3 seconds to pass him and another train of the same length moving in the opposite direction takes 4 seconds. The time taken by the trains to pass each other will be

Accepted Answer

Concept & Logic: When two trains of equal length cross a stationary point (a man), their speeds are inversely proportional to the time taken. When crossing each other in opposite directions, the total distance is the sum of their lengths, and the relative speed is the sum of their individual speeds. Speed = \frac{Distance}{Time} Step-by-Step Solution: 1. Define Length and Speeds: - Let the length of each train be L. - Speed of the first train (v_1) = L / 3. - Speed of the second train (v_2) = L / 4. 2. Calculate Relative Speed: - Since they move in opposite directions, relative speed = v_1 + v_2 = (L/3) + (L/4). - Relative speed = (4L + 3L/12) = (7L/12). 3. Calculate Time to Cross Each Other: - Total distance = L + L = 2L. - Time = \frac{Total Distance}{Relative Speed} = (2L/\frac{7L){12}}. - Time = 2L × (12/7L) = (24/7) seconds. - Converting to a mixed fraction: (24/7) = 3 (3/7) seconds. Exam Strategy & Shortcut: If two trains of equal length take t_1 and t_2 seconds to cross a stationary pole/man, the time taken to cross each other in opposite directions is given by the formula: Time = (2 × t_1 × t_2/t_1 + t_2) Substitute the values: (2 × 3 × 4/3 + 4) = (24/7) = 3 (3/7) seconds. This skips all algebraic setup. Common Pitfall: Students often average the speeds or the times, incorrectly answering 3.5 seconds. Time to cross each other is a harmonic mean-related function, not an arithmetic mean. Final Answer: Therefore, the correct answer is 3 (3/7) seconds.

Question 2

A train travelling at 48 kmph completely crosses another train having half its length and travelling in opposite direction at 42 kmph, in 12 seconds. It also passes a railway platform in 45 seconds. The length of the platform is

Accepted Answer

Concept & Logic: This problem requires applying the relative speed concept twice. First, to find the lengths of the trains by setting up their relative crossing in opposite directions. Second, using the found length of the first train to determine the length of the platform it crosses. Distance = Speed × Time Step-by-Step Solution: 1. Convert Speeds to m/s: - Speed of first train = 48 kmph = 48 × (5/18) = (40/3) m/s. - Speed of second train = 42 kmph = 42 × (5/18) = (35/3) m/s. - Relative speed (opposite directions) = 48 + 42 = 90 kmph = 90 × (5/18) = 25 m/s. 2. Find Lengths of Trains: - Let the length of the first train be 2x. The length of the second train is x. - Total distance = 2x + x = 3x. - 3x = Relative Speed × Time = 25 × 12 = 300 m. - x = 100 m. So, the first train's length is 200 m. 3. Find Length of the Platform: - The first train passes a platform of length P in 45 seconds. - Total distance = 200 + P. - Speed of first train = (40/3) m/s. - 200 + P = (40/3) × 45. - 200 + P = 40 × 15 = 600. - P = 600 - 200 = 400 m. Exam Strategy & Shortcut: Perform quick mental conversions for kmph to m/s (90 kmph is trivially 25 m/s). Establish the distance 3x = 300, so 2x = 200. For the platform, 48 kmph over 45 seconds is 48 × (5/18) × 45 = 600 m total distance. Subtract the train's length (600 - 200 = 400). Common Pitfall: A frequent error is assigning the first train length as x and the second as x/2, which introduces fractions. Using 2x and x keeps calculations clean. Additionally, remember to subtract the train's length from the total distance to find the platform length. Final Answer: Therefore, the correct answer is 400 m.

Question 3

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

Accepted Answer

Concept & Logic: When trains cross a stationary man, the time taken is proportional to their lengths relative to their own speeds (L = v × t). When crossing each other, the total time depends on the sum of their lengths divided by their relative speeds. We can use alligation to find the ratio of their speeds directly. Speed Ratio = \frac{t_{mix} - t_2}{t_1 - t_{mix}} Step-by-Step Solution: 1. Traditional Setup: - Let speeds be v_1 and v_2. - Length of train 1, L_1 = 27 v_1. - Length of train 2, L_2 = 17 v_2. 2. Crossing Each Other: - Time to cross each other = 23 seconds. - (L_1 + L_2/v_1 + v_2) = 23. - Substitute the lengths: (27 v_1 + 17 v_2/v_1 + v_2) = 23. 3. Solve for Ratio: - 27 v_1 + 17 v_2 = 23 v_1 + 23 v_2. - 27 v_1 - 23 v_1 = 23 v_2 - 17 v_2. - 4 v_1 = 6 v_2. - (v_1/v_2) = (6/4) = (3/2). Exam Strategy & Shortcut: Use the Rule of Alligation for a lightning-fast solution. - Time 1 = 27 - Time 2 = 17 - Mean Time = 23 Ratio of speeds = (23 - 17) : (27 - 23) = 6 : 4 = 3 : 2. Common Pitfall: A common mistake is trying to calculate the lengths of the trains without recognizing that the speeds and lengths are interdependent, resulting in a perceived lack of data. Final Answer: Therefore, the correct answer is 3 : 2.

Question 4

Two trains, 130 and 110 metres long, are going in the same direction. The faster train takes one minute to pass the other completely. If they are moving in opposite directions, they pass each other completely in 3 seconds. Find the speed of the fast…

Accepted Answer

Concept & Logic: The total distance covered when trains pass each other is the sum of their lengths, regardless of direction. In the same direction, relative speed = v_1 - v_2. In opposite directions, relative speed = v_1 + v_2. Relative Speed = \frac{Total Distance}{Time} Step-by-Step Solution: 1. Find Total Distance: - Length of first train = 130 m. - Length of second train = 110 m. - Total Distance = 130 + 110 = 240 m. 2. Calculate Relative Speeds: - Same direction (Time = 60 s): v_1 - v_2 = (240/60) = 4 m/s. - Opposite directions (Time = 3 s): v_1 + v_2 = (240/3) = 80 m/s. 3. Solve for Faster Train Speed (v_1): - Add the two equations: - (v_1 - v_2) + (v_1 + v_2) = 4 + 80. - 2v_1 = 84. - v_1 = 42 m/s. Exam Strategy & Shortcut: Once you have the relative speeds (4 and 80), the speed of the faster entity is simply the average of the two relative speeds: (80 + 4/2) = 42 m/s. The speed of the slower entity would be (80 - 4/2) = 38 m/s. Common Pitfall: Forgetting to convert "one minute" into 60 seconds, leading to drastically incorrect values for the relative speed in the same direction. Final Answer: Therefore, the correct answer is 42 m/sec.

Question 5

A train which is moving at an average speed of 40km/h reaches its destination on time. When its average speed reduces to 35 km/h, then it reaches its destination 15 minutes late. The distance travelled by the train, is [CLAT Exam, 2016]

Accepted Answer

Concept & Formula: When a fixed distance is covered at two different speeds resulting in a time difference \Delta t, the distance D can be found using the formula: D = (S_1 × S_2/S_1 - S_2) × \Delta t Step-by-Step Solution: 1. Given Data: - Speed 1 (S_1) = 40 km/h. - Speed 2 (S_2) = 35 km/h. - Time difference (\Delta t) = 15 minutes = (15/60) = (1/4) hours. 2. Apply Distance Formula: - D = (40 × 35/40 - 35) × (1/4) - D = (1400/5) × (1/4) - D = 280 × (1/4) = 70 km. Exam Strategy & Shortcut: Memorize the product-of-speeds formula for constant distance problems. It bypasses setting up complex algebraic fractions entirely and yields the answer in seconds. Common Pitfall: Forgetting to convert the time difference from minutes to hours ((15/60)) before applying it to speeds given in km/h. Final Answer: Therefore, the correct answer is 70km.

Question 6

A train takes 9 sec to cross a pole: If the speed of the train is 48 kmph, then length of the train is [Indian Railway Group 'D' Exam, 2014]

Accepted Answer

Concept & Formula: When a train crosses a pole (a point object), the distance covered is equal to the length of the train itself. Convert speed from kmph to m/s by multiplying by (5/18). Length = Speed × Time Step-by-Step Solution: 1. Convert Speed: - Speed = 48 kmph = 48 × (5/18) = (240/18) = (40/3) m/s. 2. Calculate Length: - Time = 9 seconds. - Length = (40/3) × 9 = 40 × 3 = 120 m. Exam Strategy & Shortcut: Simplify fractions early. Notice that 48 × (5/18) simplifies by dividing by 6 ((8 × 5/3) = (40/3)), which then easily multiplies with 9 to give 40 × 3 = 120. Common Pitfall: Forgetting to convert kmph to m/s, which results in multiplying 48 by 9 directly and getting completely incorrect magnitudes. Final Answer: Therefore, the correct answer is 120 m.

Question 7

A train passes two bridges of lengths 500 m and 250 m in 100 seconds and 60 seconds respectively. The length of the train is [SSC—CHSL (10+2) Exam, 2015]

Accepted Answer

Concept & Logic: The speed of a train remains constant whether it crosses one bridge or another. The speed is equal to the total distance (train length + bridge length) divided by the time taken. Speed = \frac{L + Bridge_1}{t_1} = \frac{L + Bridge_2}{t_2} Step-by-Step Solution: 1. Set Up Equation: - Let train length be L. - (L + 500/100) = (L + 250/60) 2. Simplify and Solve: - Cross-multiply: 60(L + 500) = 100(L + 250) - Divide by 20 to simplify: 3(L + 500) = 5(L + 250) - 3L + 1500 = 5L + 1250 - 2L = 250 - L = 125 m. Exam Strategy & Shortcut: Reduce the time ratio first (100 : 60 = 5 : 3). Since speeds are equal, the distance ratios must also balance, making quick work of the linear equation. Common Pitfall: Forgetting to add the train's length (L) to both bridge lengths when setting up the speed equivalence, treating the bridges as standalone distances. Final Answer: Therefore, the correct answer is 125 m.

Question 8

Train A travelling at 63 kmph can cross a platform 199.5 m long in 21 seconds. How much time would train A take to completely cross (from the moment they meet) train B, 157m long and travelling at 54 kmph in opposite direction which train A is trave…

Accepted Answer

Concept & Logic: First, find the length of Train A using its speed and time taken to cross the platform. Then, calculate the relative speed of both trains moving in opposite directions to find the total crossing time. Time = \frac{Length of A + Length of B}{Speed of A + Speed of B} Step-by-Step Solution: 1. Convert Speed of Train A: - Speed of A = 63 kmph = 63 × (5/18) = (35/2) = 17.5 m/s. 2. Find Length of Train A (L_A): - Total distance in 21 seconds = 17.5 × 21 = 367.5 m. - L_A + 199.5 = 367.5 \Rightarrow L_A = 168 m. 3. Calculate Relative Speed: - Speed of B = 54 kmph = 54 × (5/18) = 15 m/s. - Relative speed = 17.5 + 15 = 32.5 m/s. 4. Find Crossing Time: - Total distance = L_A + L_B = 168 + 157 = 325 m. - Time = (325/32.5) = 10 seconds. Exam Strategy & Shortcut: Convert all speeds cleanly into decimals or fractions (e.g., 17.5 and 15) to make relative speed addition (32.5) and division with 325 exceptionally straightforward without calculator reliance. Common Pitfall: Forgetting to add the two train lengths together when crossing each other, or mixing up the relative speed signs for opposite directions. Final Answer: Therefore, the correct answer is 10.

Question 9

A train running at a certain speed crosses a stationary engine in 20 seconds. To find out the speed of the train, which of the following information is necessary?

Accepted Answer

Concept & Distance Formula: When a train crosses a stationary object that has a significant length (like a platform, bridge, or another engine), the total distance covered is the sum of the lengths of both the train and the object. Speed = (Length\ of\ Train + Length\ of\ Object/Time\ Taken) Step-by-Step Solution: • Given: The time taken to cross the stationary engine = 20 seconds. • Let the length of the train be L_1 and the length of the stationary engine be L_2. • The total distance to be covered by the train to completely clear the engine = L_1 + L_2. • Using the speed formula: Speed = (L_1 + L_2/20) • Since the formula requires both L_1 and L_2 to calculate a numerical value for the speed, we need both the length of the train and the length of the engine. Exam Strategy & Shortcut: For problems involving a train crossing a non-point stationary object, the "distance" is always the sum of both lengths. If either length is missing, you cannot find the speed. You can immediately mark "both are needed" without writing down the equation. Common Pitfall: A frequent error is treating the stationary engine like a pole or a standing man, assuming its length is zero. Always check if the crossed object has a measurable length. Final Answer: Therefore, the correct answer is Both the length of the train and the length of the engine.

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