Topics · Primary Maths · Rate & Speed
Rate and speed: how much per one
A rate is how much of one thing happens per one of another. Speed is the rate that gets its own name — distance per unit of time — and the same triangle of three quantities covers both.
- 101 exam papers analysed
- appears in 57% of P6 Maths papers
- ~3 min read
01 Per one
A rate compares two different quantities: $3 per kilogram, 5 pages per minute, 120 words per page. The word per means for each one.
The move is always the same as a bar model: get to one of something, then multiply. If 4 kg cost $12, one kilogram costs $3, and 7 kg cost $21.
Speed is a rate where the second quantity is time. 60 km/h means 60 kilometres for each hour.
02 The three quantities
Distance, speed and time. Given any two, the third follows, and the three arrangements are the same relationship rewritten.
- Distance = speed × time. 60 km/h for 2 hours covers 120 km.
- Speed = distance ÷ time. 120 km in 2 hours is 60 km/h.
- Time = distance ÷ speed. 120 km at 60 km/h takes 2 hours.
- Check the units match before dividing. A speed in km/h needs the time in hours — 30 minutes is 0.5 h, not 30.
03 Where the marks go: units
The arithmetic on this topic is easy and the units are not. A speed in km/h with a time in minutes gives an answer that is wrong by a factor of 60, and the working looks perfectly tidy.
Convert first, calculate second. Put everything into the units the answer needs before touching the numbers.
04 Average speed over a whole journey
Average speed = total distance ÷ total time. Not the average of the two speeds — that is only right when the two parts take the same time, which they usually do not.
Someone driving 60 km at 60 km/h and then 60 km at 30 km/h takes 1 hour and then 2 hours: 120 km in 3 hours, so an average of 40 km/h. Averaging 60 and 30 would give 45, and is wrong.
05 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- ratemore than onerates
How much of one quantity there is for each one of another, such as $3 per kilogram.
- speedmore than onespeeds
A rate of distance per unit of time, such as 60 km/h.
- per
For each one. 60 km per hour means 60 km for each hour.
- average speed
Total distance divided by total time — not the average of the separate speeds.
- km/hsaykil-uh-mee-tuhz puh OWuh
Kilometres per hour, the usual unit of speed in these questions.
06 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P5
Rate as an amount per one, and simple cost and quantity problems.
Practise P5 Rate & Speed — 10 questions →17% of P5 papers
- P6
Speed, distance and time, unit conversions inside them, and average speed over a whole journey.
Practise P6 Rate & Speed — 10 questions →57% of P6 papers
07 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“90 km at 60 km/h takes 1 hour 50 minutes.”
90 ÷ 60 = 1.5 hours, and half an hour is 30 minutes. 1 h 30 min.
“He went 60 km/h then 30 km/h, so his average was 45 km/h.”
Total distance ÷ total time. 120 km in 3 hours is 40 km/h. Averaging the speeds only works if the two parts take equal time.
“Speed 50 km/h, time 30 minutes, so distance = 50 × 30 = 1500 km.”
The units must agree. 30 minutes is 0.5 h, so the distance is 25 km.
“Time = speed ÷ distance.”
The other way round: time = distance ÷ speed. Check with a case you know — 120 km at 60 km/h must take 2 hours.
“The answer is 1.5, so I wrote 1.5 hours and stopped.”
If the question asked in hours and minutes, 1.5 is not the answer it wanted. Read the last line.
Now try it
20 questions across 2 levels, marked as you go with the working shown.