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Thinking in Rates

Pace, speed and the arithmetic of a race: one number over another, the same speed in two units, why a longer race is slower per mile, and how age-grading lets a family compare across the generations.

Spark & Whetstone · sparkwhetstone.org/learn/thinking-in-rates · CC BY-NC-SA 4.0. Answers are at the end, so you can try each question first.

Lesson 1. One number over another

A rate is one quantity divided by another. Pace is time over distance: 30 minutes for 5 kilometres is 6 minutes per kilometre.

Speed is the same pair the other way up, distance over time. Neither is better; they answer different questions. Pace tells a runner how a lap should feel; speed tells a driver how far an hour goes.

Question. You walk 8 kilometres in 40 minutes. What is your pace?

  1. 5 minutes per kilometre
  2. 8 minutes per kilometre
  3. 0.2 kilometres per hour

Say it back. In your own words: what is the difference between pace and speed?

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Lesson 2. The same speed, two ways

Converting between speed and pace is turning the fraction over and fixing the units. 12 kilometres an hour means 60 minutes for 12 kilometres, which is 5 minutes per kilometre.

A treadmill shows speed and a running watch shows pace, so the same run can look like two different numbers. Turning one into the other is a single division.

Question. A treadmill is set to 10 kilometres an hour. What pace is that?

  1. 6 minutes per kilometre
  2. 10 minutes per kilometre
  3. 4 minutes per kilometre

Say it back. Say it back: how do you turn a speed into a pace?

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Lesson 3. A longer race is slower

Nobody holds the same pace for ten kilometres as for five. In 1977 the engineer Peter Riegel noticed that race times grow a little faster than distance: roughly, time multiplies by the distance ratio raised to the power 1.06.

Doubling the distance therefore multiplies the time by about 2.08, not 2. A 30-minute 5 km suggests a 10 km of about 62 or 63 minutes, not 60.

It is a rough rule fitted to trained runners, and it gets worse the further you stretch it. Use it to pick a sensible first pace, not to make a promise.

Question. Using the rough rule, a 30-minute 5 km suggests a 10 km of about how long?

  1. About 62 to 63 minutes
  2. Exactly 60 minutes
  3. About 75 minutes

Say it back. In your own words: why should your pace be a little slower in a longer race?

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Lesson 4. Age-grading

Age-grading compares a time with the best recorded time for someone of the same age and sex at that distance, and gives a percentage. 70% means seventy per cent as fast as that best.

It lets a grandmother, a father and a teenager compare runs fairly, because each is measured against the best at their own stage of life rather than against each other.

An age-graded score is a way to compare. It is not a claim about health, and a low number is not a verdict on anyone.

Question. What does an age-graded score of 70% mean?

  1. The run was 70% as fast as the best time recorded at that age and distance.
  2. The runner beat 70% of the people in the race.
  3. The runner is 70% fitter than average.

Say it back. Say it back: why can age-grading make a family race fair across generations?

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At the table: The family relay

Players: 2 to 8, any ages, walkers welcome. You'll need: a measured loop (a park path, a block), a phone or watch with a timer, paper.

  1. Everyone walks or runs one lap and writes down their time.
  2. Each person works out their own pace for the lap, then predicts the whole relay's time if everyone does one lap in turn.
  3. Run the relay. Compare the prediction with the real total and talk about which leg surprised you.

Answers

  1. 5 minutes per kilometre — 40 minutes over 8 kilometres is 40 ÷ 8 minutes for each one.
  2. 6 minutes per kilometre — 60 minutes for 10 kilometres is 60 ÷ 10 minutes for each kilometre.
  3. About 62 to 63 minutes — 30 × 2.08 is about 62.5 minutes: a little slower per kilometre than the 5 km.
  4. The run was 70% as fast as the best time recorded at that age and distance. — The percentage is your time against the best recorded at the same age, sex and distance. It says nothing about the other people in the race.