Speed, distance and time are linked by speed = distance ÷ time. At GCSE you must rearrange this formula fluently, convert between units, and tackle multi-step problems — including finding the average speed for a whole journey with two stages. Most errors come from unit mismatches or misapplying the average-speed formula, so careful working is essential.
What is the speed–distance–time formula?
The three quantities are related by:
speed = distance ÷ time (abbreviated S = D/T)
Rearranged:
- distance = speed × time (D = S × T)
- time = distance ÷ speed (T = D/S)
A useful memory aid is the formula triangle: write D at the top, S and T at the bottom. Cover the quantity you want to find; the remaining two show what to do (side-by-side means multiply; top over bottom means divide).
How do you rearrange the formula?
If you know any two of the three quantities, rearrange to find the third.
| Known values | Formula to use | Example |
|---|---|---|
| Distance and time | S = D ÷ T | 150 km in 2.5 h → S = 150 ÷ 2.5 = 60 km/h |
| Speed and time | D = S × T | 80 km/h for 3.5 h → D = 80 × 3.5 = 280 km |
| Distance and speed | T = D ÷ S | 210 km at 70 km/h → T = 210 ÷ 70 = 3 h |
How do you convert units within the formula?
The units of speed follow automatically from the units of distance and time.
Speed in km/h: use distance in km and time in hours. Speed in m/s: use distance in metres and time in seconds.
Converting between km/h and m/s:
- To go from km/h to m/s: ÷ 3.6 (because 1 km = 1000 m and 1 hour = 3600 s, so 1000/3600 = 1/3.6)
- To go from m/s to km/h: × 3.6
Example: convert 90 km/h to m/s. 90 ÷ 3.6 = 25 m/s
Example: a car travels at 54 km/h. How far does it travel in 40 minutes?
- Convert time to hours: 40 min = 40/60 h = 2/3 h.
- Distance = 54 × 2/3 = 36 km.
Always express time in the same unit as the speed's denominator. A common error is calculating 54 × 40 = 2160 (treating minutes as hours).
What is average speed and why is it tricky?
Average speed for a whole journey is:
average speed = total distance ÷ total time
It is NOT the mean of the individual speeds, because sections of a journey can take very different amounts of time.
Worked example: a cyclist rides 30 km at 15 km/h, then a further 20 km at 10 km/h. Find the average speed for the whole journey.
| Leg | Distance | Speed | Time |
|---|---|---|---|
| First | 30 km | 15 km/h | 30 ÷ 15 = 2 h |
| Second | 20 km | 10 km/h | 20 ÷ 10 = 2 h |
| Total | 50 km | — | 4 h |
Average speed = 50 ÷ 4 = 12.5 km/h
(Not (15 + 10) ÷ 2 = 12.5 here — coincidental agreement! In general, the arithmetic mean of the speeds is wrong.)
Why is the arithmetic mean wrong? The cyclist spent 2 hours at each speed, so in this case the mean happens to be correct — but if the legs had different durations, the time-weighted average speed would differ from the arithmetic mean.
How do you use distance–time graphs?
On a distance–time graph:
- The gradient of the line at any point gives the speed at that moment.
- A horizontal line (gradient = 0) means the object is stationary.
- A steeper section means greater speed.
- The speed for a straight section = (change in distance) ÷ (change in time).
Example: a straight section rises from 20 km to 50 km over 1.5 hours. Speed = (50 − 20) ÷ 1.5 = 30 ÷ 1.5 = 20 km/h.
Frequently asked questions
What is the difference between speed and velocity?
Speed is a scalar (size only); velocity is a vector (size and direction). At GCSE maths, most questions use speed. Physics questions may use velocity and distinguish positive from negative directions. In a maths exam, treat "velocity" as speed with a direction noted.
How do I handle a question where time is given in mixed units, e.g. 2 hours 15 minutes?
Convert entirely to hours or entirely to minutes before substituting into the formula. Two hours and 15 minutes = 2.25 hours (since 15/60 = 0.25). Never substitute 2.15 — that would be 2 hours and 0.15 of an hour (9 minutes), not 15 minutes.
Does the average-speed formula ever equal the arithmetic mean of the speeds?
Yes — when the time spent at each speed is equal. If a driver goes 60 km/h for exactly 1 hour and 40 km/h for exactly 1 hour, average speed = (60 + 40) ÷ 2 = 50 km/h. But if the distances (not times) are equal, the correct formula gives a different answer: travelling 60 km at 60 km/h takes 1 h; travelling 60 km at 40 km/h takes 1.5 h. Average speed = 120 ÷ 2.5 = 48 km/h.
Can I use the formula triangle without understanding the algebra?
The triangle helps in straightforward one-step problems, but multi-step questions require you to apply the formula correctly in context. Always write the formula explicitly (S = D/T) before substituting — this avoids errors when the unknown is D or T rather than S.
For GCSE maths problem-solving practice with Professor Pi, visit aitutors.me.