Speed, distance and time: the complete method

One formula, three unit rules and the average-speed trap. A step-by-step guide to speed, distance and time questions, including trains and relative speed.

Speed, distance and time questions look varied: cars, trains, people walking, two vehicles meeting. Underneath, they all come from one relationship:

Distance = Speed × Time

Rearranged, Speed = Distance ÷ Time and Time = Distance ÷ Speed. If you can see which two of the three a question gives you, you can always find the third.

Step 1: get the units right

Most mistakes in this topic are unit mistakes, not maths mistakes. Before calculating anything, make sure speed, distance and time use matching units.

• To turn km/h into m/s, multiply by 5/18. (1 km = 1,000 m and 1 hour = 3,600 s, and 1,000 ÷ 3,600 = 5/18.)
• To turn m/s into km/h, multiply by 18/5.
• Turn minutes into hours by dividing by 60: 45 minutes = 0.75 hours.

Example: 72 km/h in m/s is 72 × 5/18 = 20 m/s.

Step 2: the basic questions

Example: A car travels at 60 km/h for 2 hours 30 minutes. How far does it go?

• Time = 2.5 hours.
• Distance = 60 × 2.5 = 150 km.

Example: A cyclist covers 18 km in 45 minutes. What is her speed in km/h?

• Time = 45 ÷ 60 = 0.75 hours.
• Speed = 18 ÷ 0.75 = 24 km/h.

The average speed trap

Suppose you drive to a town at 20 km/h and come back along the same road at 60 km/h. Your average speed is not (20 + 60) ÷ 2 = 40 km/h.

Average speed is always total distance ÷ total time. Say the town is 60 km away:

• Going takes 60 ÷ 20 = 3 hours. Coming back takes 60 ÷ 60 = 1 hour.
• Total distance = 120 km, total time = 4 hours.
• Average speed = 120 ÷ 4 = 30 km/h.

When the two distances are equal, there is a shortcut: average speed = 2ab ÷ (a + b). Here, 2 × 20 × 60 ÷ 80 = 30 km/h. Notice the average is pulled towards the slower speed, because you spend more time going slowly.

Trains

A train has length, and that changes what "crossing" means.

• Crossing a pole or a person standing still: the train covers its own length.
• Crossing a platform or a bridge: the train covers its length plus the platform's length.

Example: A 150 m train crosses a pole in 15 seconds. Its speed is 150 ÷ 15 = 10 m/s, which is 10 × 18/5 = 36 km/h.

Example: The same train crosses a 250 m platform. It covers 150 + 250 = 400 m at 10 m/s, so it takes 40 seconds.

Relative speed

When two things move at the same time, use their relative speed:

• Moving towards each other (or in opposite directions): add the speeds.
• Moving in the same direction: subtract the smaller speed from the larger.

Example: Two people start 30 km apart and walk towards each other at 4 km/h and 6 km/h. Their gap closes at 4 + 6 = 10 km/h, so they meet after 30 ÷ 10 = 3 hours.

Example: A car at 70 km/h chases a bus 10 km ahead travelling at 50 km/h. The gap closes at 70 − 50 = 20 km/h, so the car catches up in 10 ÷ 20 = half an hour.

A checklist for any question

1. Write down which two of speed, distance and time are given.
2. Convert everything to matching units.
3. For trains, decide what length is being covered.
4. For two movers, decide whether to add or subtract speeds.
5. For averages, use total distance ÷ total time.

Practise

Try the solved speed, distance and time examples. Time and work questions use a very similar "rate × time" idea, so they make good practice too.

Practise it: solved examples.