A man is standing on a railway bridge which is 110 m long. He finds that a train crosses the bridge in 30 seconds but himself in 18 seconds. Find the length of the train and its speed.

Let's solve this step by step:

Step 1: First, we need to find the speed of the train when it crosses the man. The speed of an object is calculated by dividing the distance it travels by the time it takes. In this case, the distance is the length of the train (which we don't know yet), and the time is 18 seconds. So, we can represent the speed of the train (v) as v = Length of the train / 18.

Step 2: Next, we need to find the speed of the train when it crosses the bridge. This time, the distance is the length of the bridge plus the length of the train (110m + Length of the train), and the time is 30 seconds. So, we can represent this speed as v = (110m + Length of the train) / 30.

Step 3: Since the speed of the train is the same in both situations, we can set the two equations equal to each other and solve for the length of the train:

Length of the train / 18 = (110m + Length of the train) / 30.

Step 4: Cross-multiply to get rid of the fractions:

30 * Length of the train = 18 * (110m + Length of the train).

Step 5: Distribute the 18:

30 * Length of the train = 1980m + 18 * Length of the train.

Step 6: Subtract 18 * Length of the train from both sides:

12 * Length of the train = 1980m.

Step 7: Divide by 12 to solve for the length of the train:

Length of the train = 1980m / 12 = 165m.

So, the length of the train is 165 meters.

Step 8: To find the speed of the train, substitute the length of the train into the equation from Step 1:

v = Length of the train / 18 = 165m / 18 = 9.17 m/s.

So, the speed of the train is 9.17 meters per second.