Quantitative Apptitude - Problems On Trains Online Test 3

Explanation
The ratio of the speeds of the two trains can be determined by using the concept of relative speed. When the two trains are moving in opposite directions, their speeds are added. When they cross each other, their speeds are subtracted. Let's assume the speeds of the two trains are 3x and 2x respectively. When they are moving in opposite directions, the relative speed is 3x + 2x = 5x. The time taken to cross the man is 27 seconds, so the distance covered is 27 * 5x = 135x. Similarly, when they cross each other, the relative speed is 3x - 2x = x. The time taken to cross each other is 23 seconds, so the distance covered is 23 * x = 23x. Since the two distances are the same, we can equate them: 135x = 23x. Solving this equation gives x = 2. Therefore, the ratio of the speeds is 3x:2x = 3:2.

Explanation
The train is running faster than the man, so it will eventually pass him. To calculate the time it takes for the train to pass the man, we need to find the relative speed between the train and the man. The relative speed is the difference between the speed of the train and the speed of the man, which is (68 kmph - 8 kmph) = 60 kmph.

We need to convert this relative speed to meters per second, so we divide by 3.6 (since 1 kmph = 1000/3600 m/s).

60 kmph / 3.6 = 16.67 m/s.

Now, we can calculate the time it takes for the train to pass the man by dividing the length of the train (150 m) by the relative speed (16.67 m/s):

150 m / 16.67 m/s ≈ 9 sec.

Therefore, the train will pass the man in approximately 9 seconds.

Explanation
The length of the train can be calculated using the formula: Length = Speed x Time. In this case, the speed of the train is given as 60 km/hr and the time taken to cross the pole is 9 seconds. To calculate the length, we need to convert the speed from km/hr to m/s, which is 60 km/hr = 60 x (1000/3600) m/s = 16.67 m/s. Plugging in the values, Length = 16.67 m/s x 9 s = 150 m. Therefore, the length of the train is 150 m.

Explanation
The time taken by the train to cross a tree can be calculated by dividing the distance traveled by the train (150 m) by its speed (90 km/h). First, we convert the speed from km/h to m/s by dividing it by 3.6 (1 km/h = 0.2778 m/s). So, the speed of the train is 25 m/s. Dividing the distance (150 m) by the speed (25 m/s) gives us the time taken, which is 6 seconds.

Explanation
The length of the platform can be determined by subtracting the distance covered by the train in 20 seconds from the distance covered by the train in 36 seconds. The train covers a distance of 54 km/hr in 36 seconds, which is equal to 54000/3600 = 15 m/s. Therefore, the train covers a distance of 15 m/s x 36 s = 540 m in 36 seconds. Similarly, the train covers a distance of 15 m/s x 20 s = 300 m in 20 seconds. Subtracting the distance covered in 20 seconds from the distance covered in 36 seconds gives us the length of the platform, which is 540 m - 300 m = 240 m.

Explanation
To convert the speed from kilometers per hour to meters per second, we need to multiply it by 1000/3600. In this case, the train's speed of 108 kmph would be equal to (108 * 1000) / 3600 = 30 meters per second.

Explanation
The time taken by the train to pass the man can be calculated using the formula: time = distance/speed. The distance the train needs to cover to pass the man is equal to its own length, which is 100 m. The speed of the train is given as 30 km/hr. To convert this to m/s, we divide by 3.6 (1 km/hr = 1000 m/3600 s). Therefore, the speed of the train is 8.33 m/s. Substituting these values into the formula, we get time = 100 m / 8.33 m/s = 12 sec.

Explanation
The man in the slower train passes the faster train in 8 seconds, which means that in 8 seconds, the faster train travels a distance equal to its own length plus the length of the slower train. Since the slower train is traveling at 36 kmph and the faster train is traveling at 45 kmph, the relative speed between the two trains is 36 + 45 = 81 kmph. Converting this to meters per second gives us 81 * (5/18) = 45/2 m/s. Therefore, in 8 seconds, the faster train travels a distance of (45/2) * 8 = 180 meters. Hence, the length of the faster train is 180 meters.

Explanation
The train is moving in the opposite direction of the man, so their relative speed is the sum of their speeds, which is (59 kmph + 7 kmph) = 66 kmph. We need to convert this speed to meters per second, so we divide by 3.6: (66 kmph / 3.6) = 18.33 m/s. The time taken to pass the man is equal to the distance traveled by the train, which is 220 m, divided by the relative speed: (220 m / 18.33 m/s) = 12 sec.