MISSION · Number sense
Speed, Distance, and Time
A rate problem you already meet every day
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
Speed is distance per unit of time, and the three quantities are locked together so tightly that knowing any two gives the third.
\[\text{speed} = \frac{\text{distance}}{\text{time}}\]The three arrangements
\[s = \frac{d}{t} \qquad d = s \times t \qquad t = \frac{d}{s}\]They are one equation rearranged, not three facts to memorise. If you can rearrange, you only ever need the first.
The formula triangle — \(d\) on top, \(s\) and \(t\) below — is the usual memory aid: cover the one you want and the triangle shows what to do with the other two. It works, but it’s a crutch. The reasoning behind it is simpler: travelling 60 km/h for 2 hours obviously covers 120 km, because each hour adds 60.
Units decide everything
A speed carries two units, and both have to match the numbers you feed it.
km/h needs distance in kilometres and time in hours. m/s needs metres and seconds. Putting minutes into a km/h calculation gives an answer wrong by a factor of 60 — and that’s the single most common error in the topic.
Converting a time like 2 hours 30 minutes is where it usually goes wrong. That’s 2.5 hours, not 2.3. Minutes convert by dividing by 60, so 30 minutes is \(\frac{30}{60} = 0.5\) of an hour. 20 minutes is \(\frac{1}{3}\) of an hour, or 0.333…
To switch a speed between units:
\[1 \text{ m/s} = 3.6 \text{ km/h}\]because a metre per second is 3600 metres an hour, which is 3.6 km.
Average speed is not the average of the speeds
This one catches nearly everybody.
Drive 60 km at 60 km/h, then 60 km at 30 km/h. The average speed is not 45 km/h.
Work it out properly. The first stretch takes 1 hour, the second takes 2 hours. That’s 120 km in 3 hours:
\[\frac{120}{3} = 40 \text{ km/h}\]The slow section takes longer, so it counts for more of the journey. Average speed is always total distance over total time, never the mean of the speeds — and the true answer is always lower than the naive average.
Distance-time graphs
Plot distance against time and the slope is the speed. A steeper line means faster travel; a horizontal line means stopped, since time passes and distance doesn’t change.
That’s the same slope-as-a-rate idea from straight-line graphs, and it’s why these graphs appear in both the science and the maths syllabus.
Common mix-ups
- Writing 2 hours 30 minutes as 2.3. It’s 2.5.
- Mismatched units — minutes with km/h, or metres with mph.
- Averaging the speeds instead of dividing total distance by total time.
- Reading a horizontal line as “not moving on the graph” rather than “stationary”.
Check yourself
A train covers 210 km in 2 hours 30 minutes. What is its average speed? How long would 350 km take at that speed?
\(2\text{h}30 = 2.5\) hours, so \(210 \div 2.5 = 84\) km/h. Then \(350 \div 84 \approx 4.17\) hours — about 4 hours 10 minutes.
LEVEL 3 · REFLECT
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