Speed, Time and Distance

Speed, time and distance questions are a favourite in placement tests because they look like everyday situations: a car trip, two trains passing, a boat in a river. All of them rely on one triangle of ideas, and a few special cases that you can learn in an hour.

The basic relation

Speed = Distance / Time, so Distance = Speed × Time and Time = Distance / Speed.

Keep the units consistent. If speed is in km/h, then distance must be in km and time in hours.

Unit conversion

• km/h to m/s: multiply by 5/18. So 72 km/h = 72 × 5/18 = 20 m/s.
• m/s to km/h: multiply by 18/5. So 15 m/s = 54 km/h.
• Minutes to hours: divide by 60. Forty minutes is 2/3 of an hour.

Proportion rules

• For the same distance, speed and time are inversely proportional. If speed goes up by a factor of 2, time halves. If the speeds are in the ratio a : b, the times are in the ratio b : a.
• For the same time, distance is proportional to speed.

Average speed

Average speed = Total distance / Total time. It is not the plain average of the speeds unless the times are equal.

When the same distance is covered at two speeds x and y (for example, to a place and back), the average speed is the harmonic mean:

2xy / (x + y).

Example: going at 40 km/h and returning at 60 km/h gives 2 × 40 × 60 / 100 = 48 km/h, not 50.

Relative speed

• Two objects moving in opposite directions (towards or away from each other): add the speeds.
• Two objects moving in the same direction: subtract the speeds.

Example: two cars start 300 km apart and drive towards each other at 40 and 60 km/h. The gap closes at 100 km/h, so they meet after 3 hours.

Trains

• A train passing a pole, a tree or a standing person covers its own length. Time = length / speed.
• A train passing a platform or bridge covers its own length plus the platform's length.
• Two trains crossing each other cover the sum of their lengths at their relative speed.

Example: a 150 m train at 54 km/h (= 15 m/s) crosses a pole in 150 / 15 = 10 seconds, and crosses a 100 m platform in 250 / 15 = 16.67 seconds.

Boats and streams

Let the boat's speed in still water be u and the speed of the stream be v.

• Downstream speed = u + v (the stream helps).
• Upstream speed = u − v (the stream resists).
• Therefore u = (downstream + upstream) / 2 and v = (downstream − upstream) / 2.

Example: u = 8 km/h and v = 2 km/h. Downstream speed is 10 km/h and upstream speed is 6 km/h. A 30 km trip takes 3 hours downstream and 5 hours upstream.

Worked examples

Example 1. A bus covers 150 km in 2.5 hours. Its speed is 150 / 2.5 = 60 km/h. At that speed, 210 km takes 3.5 hours.

Example 2. Ravi walks at 6 km/h for 1.5 hours and then at 4 km/h for 1 hour. Total distance = 9 + 4 = 13 km, total time = 2.5 hours, so average speed = 5.2 km/h.

Example 3. A train 120 m long passes a man running at 6 km/h in the same direction in 12 seconds. The train's speed relative to the man = 120 / 12 = 10 m/s = 36 km/h. So the train's speed is 36 + 6 = 42 km/h.

Common mistakes

• Averaging the two speeds for a round trip instead of using total distance over total time.
• Forgetting to add the lengths when a train crosses a platform or another train.
• Mixing km/h and m/s in one calculation.
• Adding speeds for same-direction movement, or subtracting them for opposite directions.

Exam tips

1. Convert all units first, then calculate.
2. For a round trip at two speeds, use 2xy / (x + y).
3. Draw a quick diagram for meeting and overtaking questions.

Take the quiz and check each answer against the reasoning shown.