Two halves of one journey
Seeing why an average of rates is not the average rate, on one journey, four times.
Scenario
A driver covers a 180 km route in two equal halves.
The first 90 km is on single-track roads, driven at an average of 30 km/h. The second 90 km is on a motorway, driven at an average of 90 km/h.
No stops are made, and each leg is driven at its stated average throughout.
Answers in order, explanations withheld, and a report on where the chain broke.
- 20.1RatesFoundation
How long does each leg take?
- A3 hours and 1 hour
- B2 hours and 2 hours
- C1 hour and 3 hours
- D3 hours and 3 hours
Show the answer and reasoningHide the reasoning
3 hours and 1 hour
Time is distance divided by speed: 90 / 30 = 3 hours on the first leg, and 90 / 90 = 1 hour on the second. The distances are equal and the times are not, which is the whole set in one line.
- 20.2RatesStandard
Builds on: The two times from question 1. Without them the total time is unknown, and averaging the speeds is the only thing left to do.
What is the average speed for the whole journey?
- A60 km/h
- B45 km/h
- C50 km/h
- D40 km/h
Show the answer and reasoningHide the reasoning
45 km/h
Average speed is total distance over total time: 180 / 4 = 45 km/h. Averaging the two speeds gives 60, which would only be right if equal time were spent at each — and question 1 showed the times are 3 hours against 1.
- 20.3Interpreting modelsChallenge
Builds on: Questions 1 and 2 together. One gives the 3-to-1 split of time and the other gives the answer it produces; this question is why those two facts belong together.
Why is the answer below the midpoint of 30 and 90?
- ABecause the second leg is shorter in distance than the first.
- BBecause three quarters of the journey time is spent at the slower speed.
- CBecause averages are always pulled towards the smaller number.
- DBecause the motorway speed was not maintained throughout.
Show the answer and reasoningHide the reasoning
Because three quarters of the journey time is spent at the slower speed.
Of the four hours, three are spent at 30 km/h and one at 90. An average over time weights the slow leg three times as heavily, which drags the result down to 45. The distances are equal; it is the time that is not.
- 20.4RatesChallenge
Builds on: Question 3, which identified time as the weighting. This changes only that weighting and nothing else, and the answer that was wrong before becomes the right one.
The driver now spends one hour on each leg instead, at the same two speeds. What is the average speed?
- A45 km/h
- B60 km/h
- C90 km/h
- D30 km/h
Show the answer and reasoningHide the reasoning
60 km/h
One hour at 30 covers 30 km and one hour at 90 covers 90 km, so 120 km in 2 hours is 60 km/h. When the time is split equally, the average of the speeds is right — which is exactly the condition the first three questions showed was missing.