Average speed is total distance ÷ total time. An instantaneous speed is the speed at one moment, like the reading on a speedometer. They are the same only when the speed never changes.
This appears in there-and-back questions, graph questions and mixed problems. It uses the graph skills from distance-time graphs and speed-time graphs.
How do you calculate an average rate correctly?
- Find every distance and add them.
- Find every time and add them. For each part, time = distance ÷ speed.
- Divide the total distance by the total time.
- Sense check. The average must lie between the slowest and fastest speeds, and it is pulled towards the speed you spent longer at.
Worked example
A driver goes 120 km to a town at 60 km/h, then returns the same 120 km at 40 km/h. Find the average speed for the round trip.
Step 1, times: outward 120 ÷ 60 = 2 h. Return 120 ÷ 40 = 3 h.
Step 2, totals: distance 120 + 120 = 240 km. Time 2 + 3 = 5 h.
Step 3, average speed: 240 ÷ 5 = 48 km/h.
Now an instant reading. A car’s distance is given by s = t², with s in metres and t in seconds.
In the first 3 s it travels 9 m, so the average speed is 9 ÷ 3 = 3 m/s.
At t = 3 s the tangent to the curve has a gradient of 6, so the speed at that instant is about 6 m/s. The car is speeding up, so the speed at the end of the interval is far above the average.
The mistake to watch for
A common slip is to average the two speeds.
Mistaken answer: (60 + 40) ÷ 2 = 50 km/h
The student treated the speeds as equally weighted. The driver spent 3 hours at 40 km/h but only 2 hours at 60 km/h.
The correction is to use the totals: 240 km ÷ 5 h = 48 km/h, which is below 50 because more time was spent at the slower speed.
Check yourself
Try these, then open each answer.
1. A walker covers 3 km at 6 km/h, then 3 km at 3 km/h. Find the average speed.
Show answer
Times: 3 ÷ 6 = 0.5 h and 3 ÷ 3 = 1 h. Total time = 1.5 h. Total distance = 6 km. Average speed = 6 ÷ 1.5 = 4 km/h.
2. A runner covers 400 m in 80 s. Find the average speed in m/s. Does this tell you the speed at 40 s?
Show answer
400 ÷ 80 = 5 m/s. It does not tell you the speed at 40 s, because the runner may have been faster or slower at different times.
3. A journey of 150 km takes 2 h 30 min, including a 20 minute stop. Find the average speed including the stop, and the average speed while moving, to 1 decimal place.
Show answer
Including the stop: 150 ÷ 2.5 = 60 km/h. Moving time = 150 − 20 = 130 min = 13/6 h. Speed = 150 ÷ 13/6 = 900/13 = 69.2 km/h (1 d.p.).
Where this leads next
Put the whole module together in the rates mixed practice set, or go back to the module page. The graph-model explorer and the non-calculator working trainer are helpful for extra checking.
If the tempting wrong answer keeps winning, our teachers look for that habit in online one-to-one Mathematics tuition.