An Average Average Speed
I drove from Cambridge to Cardiff at an average speed of 50 mph, it was heavy traffic, but I managed to drive back at an average speed of 70 mph. Why wasn't my average speed for the round trip 60mph ?
What average speed for the return journey would make the round trip average 60 mph ?
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Does it matter how far Cardiff is from Cambridge ?
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Will the overall average speed change if it was twice as far ? Half as far ? 100 miles ? One mile ?
Well done Shaan from Watford Grammar School and many others who sent in well-explained correct solutions for this problem.
I noticed that neither the distance nor the journey time were given.
Time there was d / 50
Time back was d / 70
The Average speed for the round trip was total distance divided by total time
The reason the average of the average speeds isn't the overall average speed is because the journey time isn't the same there and back . . . . but the distance is, and that's useful if instead I use the reciprocal of the rate.
Speed is a rate (the rate at which distance changes as time passes) and any rate can be considered in it's reciprocal form (the rate at which time passes per unit distance)
The distance is the same so adding the reciprocal of the speeds and then dividing by two does give me the reciprocal of the overall average speed.
and this gives the answer 75 mph too.
Why do this problem:
Averages can seem rather common-place, we think we know all about them, so this problem takes 'average speed', a concept many students at this Stage will think they know, and with a simple question probes their understanding.Possible approach :
The questions below, the support activity, and the suggested extension task indicate a route into the concept of 'average speed'.For abler students grasping that any rate has a reciprocal form and that considering that alternative form might sometimes be useful is an important insight. (Miles per hour, is the reciprocal of hours per mile, for example)
Key questions :
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What does 60mph mean ?
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Why might someone think that the average of 50mph and 70 mph was 60 mph ?
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Does it matter how far Cardiff is from Cambridge ?
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What would change if it was twice as far ? Half as far ? 100 miles ? One mile ?
Possible extension :
- Can you write a formula that connects the two average speeds (there and back) with the average speed for the journey ?
- What happens when one of the two average speeds (there or back) is extremely large or extremely small ?
Possible support :
Students that do not already have a properly grounded understanding of speed as a rate of change of distance (or displacement) over time could benefit from some practical, tangible experience.Take a marked distance along the floor or the wall for example and 'step' along it with 'finger footsteps' using a stopwatch to determine the time for the journey. A 'finger footstep' is the distance between the thumb and index finger when there is a wide 'V' between them, or from thumb to 'baby' finger like the children's hand game 'incy wincy spider'. Students can be asked to make leisurely journeys, or fastest possible, and each time work out the average speed in 'footsteps' per minute. Encourage questions and challenges as students begin to give meaning to these measures and visualise the 'journey' when given the 'speed' (footsteps per minute).