Speed distance time: the formula, the triangle, average speed
Speed is distance divided by time. Turned round, distance is speed times time, and time is distance divided by speed. Average speed for a whole journey is total distance divided by total time, never the mean of two speeds.
Speed distance time is one idea in three forms: how fast, how far, how long. Know two and the third is one sum. Below: the formula, the triangle, the unit slip and the average-speed trap. More like it is in the numerical practice hub.
Speed distance time formula: which of the three do you write down?
Speed is how far you get in one unit of time, and the other two formulas are that sentence turned round. Read the question for the quantity it hides. The hidden one is what you solve for, and the other two are on the page.
Formulas
Speed = (Distance) divided by (Time)
Distance = Speed × Time
- Speed is a rate, such as kilometres per hour (km/h).
- Distance and time have to use matching units, so km goes with hours.
Worked examples
A courier drives 180 km to a depot in 3 hours. How fast was the van going, on average?
- Divide distance by time: 180 ÷ 3 = 60 km/h
A delivery run is 150 km and the van holds 50 km/h. How long does it take?
Multiplying 150 by 50 gives 7500, a run of 7500 hours. When a time comes out absurd, you multiplied where the sentence needed a division.
- Turn the formula round: time is distance ÷ speed, so 150 ÷ 50 = 3 hours
The van averaged 60 km/h, and the second run takes 3 hours.
Check it: 60 km/h for 3 hours is 180 km, the distance you started with.
Speed distance time triangle: how do you read it?
Draw a triangle with distance on top, and speed and time side by side underneath. Cover the one you want and what is left is the sum: a stack means divide, a pair side by side means multiply.
- Cover speed. Distance sits over time, so 120 km in 2 hours is 120 ÷ 2 = 60 km/h.
- Cover distance. Speed and time sit side by side, so 60 km/h for 2 hours is 60 × 2 = 120 km.
- Cover time. Distance sits over speed, so 120 km at 60 km/h is 120 ÷ 60 = 2 hours.
- What the triangle does not know. It stores the arrangement and not the units. Put 90 minutes straight in and 60 × 90 = 5400 comes out, which is not kilometres in any journey.
Minutes, hours, metres, kilometres: what do you fix before dividing?
A speed carries its units in its name. Kilometres per hour wants kilometres and hours, and minutes slipped in make the answer wrong by a factor of sixty without a hint in the arithmetic. Convert first, then divide.
Formulas
Hours = (Minutes) divided by (60)
m/s = (km/h) divided by (3.6)
- The 3.6 comes from 3600 seconds in an hour over 1000 metres in a kilometre.
- Miles against kilometres is a different conversion, and a test that mixes them gives you the factor.
Worked examples
A train covers 90 km in 45 minutes. What is its speed in km/h?
Dividing 90 by 45 gives 2, which is kilometres per minute. The question asked for kilometres per hour.
- Minutes to hours: 45 ÷ 60 = 0.75 hours
- Divide distance by time: 90 ÷ 0.75 = 120 km/h
A cyclist holds 36 km/h. How many metres per second is that?
- Divide by 3.6: 36 ÷ 3.6 = 10 m/s
The train runs at 120 km/h and the cyclist at 10 m/s.
Check it: 2 km a minute for 60 minutes is 120 km in an hour, and 10 m/s for 3600 seconds is 36000 metres, which is 36 km.
How to calculate average speed: why 40 and 60 do not make 50
Average speed is total distance divided by total time, and nothing else. The mean of two speeds is only right when you spent the same time at each. On a there-and-back trip you cover the same distance at each, so you spend longer at the slow speed and it drags the average down.
Formula
Speed = (All distance) divided by (All time)
- All distance counts every leg.
- All time is each leg's distance ÷ its speed, added up.
Worked example
A sales rep drives 120 km to a client at 40 km/h and comes back the same 120 km at 60 km/h. What is the average speed for the round trip?
Adding 40 and 60 gives 100, and half of that is 50. The slow leg lasted 3 hours against 2 for the fast one, so 50 cannot be right.
- Time out: 120 ÷ 40 = 3 hours
- Time back: 120 ÷ 60 = 2 hours
- Total time: 3 + 2 = 5 hours
- Divide the 240 km by it: 240 ÷ 5 = 48 km/h
The average speed is 48 km/h.
Check it: 48 sits below 50 because more of the time went at 40 km/h. Spend an hour at each speed instead and you cover 100 km in 2 hours, which is 50 km/h, so the simple mean works only when the times are equal.
Speed distance time questions: where do the marks go?
In a timed test the sum is easy and the set-up is where marks go. Before you calculate, name the missing quantity, check every time for minutes, and ask whether a stop counts as part of the trip.
A multi-leg question adds a trap. A break is time on the clock, so a whole-trip average counts it unless the question says it wants moving speed.
Formulas
Speed = (All distance) divided by (All time)
All time = Driving + stops
- Work each leg's time on its own, then add the break.
Worked example
A courier drives 60 km at 30 km/h, stops for a 30 minute break, then drives 90 km at 45 km/h. What is the average speed for the whole trip?
Leaving the break out gives 150 ÷ 4 = 37.5 km/h. That is the speed of the driving, and the question asked for the trip.
- Driving times: 60 ÷ 30 = 2 hours, and 90 ÷ 45 = 2 hours
- The break in hours: 30 ÷ 60 = 0.5 hours
- Total time: 2 + 2 + 0.5 = 4.5 hours
- Divide the 150 km by it: 150 ÷ 4.5 = 33.33 km/h
The average speed for the trip is 33.33 km/h.
Check it: the trip average has to land below 37.5, which is what the same drive would score with no break.
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
Which other method does your question need?
- One speed given as a rise or a fall on another: percentages, step by step
- Every numerical formula, one worked example each: the formulas reference
Want to try these under a clock? The button below opens the numerical practice page. Its 12-question diagnostic runs on a 12-minute clock, and the clock starts only when you press start on that page.
How we write, check and mark the practice questions is set out on the Methodology page.