# @stem_antics on Instagram

- **Type:** Video
- **Original URL:** https://www.instagram.com/p/DeQUe64gSAz
- **Gondola URL:** https://gondola.cc/posts/71427938-stem-antics-instagram
- **Thumbnail:** https://img.gondola.cc/tr:w-,h-,fo-auto/postThumbnails/11da4de96d.jpg
- **Posted:** 2026-10-09T01:27:26.000+00:00
- **Account Owner:** Stem Antics (@stem_antics) — https://gondola.cc/stem_antics

## Caption

A car drives 40 mph to a destination and 60 mph back. The average speed is not 50 mph. It is 48 mph.

The error is intuitive and almost universal. The arithmetic mean weights each speed equally, but the car does not spend equal time at each speed. It covers equal distance. The outbound leg takes 50 per cent longer than the return, so the slower speed dominates the clock.

Work it through for a 120-mile one-way trip:

- Outbound: 120 / 40 = 3 hours
- Return: 120 / 60 = 2 hours
- Total: 240 miles in 5 hours = 48 mph

The distance cancels entirely. For two equal-distance legs, average speed is 2ab / (a + b), the harmonic mean: the reciprocal of the average of the reciprocals. Speed is distance per time. When distance is held fixed, the reciprocal, hours per mile, is the quantity that actually adds linearly.

This generalises into one of the cleanest inequalities in mathematics. For any set of positive numbers:

- Arithmetic mean ≥ geometric mean ≥ harmonic mean
- Equality holds only when every value is identical
- For 40 and 60: AM = 50, GM ≈ 48.99, HM = 48

For two values, the three are locked together. The geometric mean squared equals the arithmetic mean multiplied by the harmonic mean: 2400 = 50 × 48.

The name is older than algebra. Archytas of Tarentum, around 400 BCE, renamed the “subcontrary” mean the harmonic mean because of its role in musical tuning. Take string lengths of 12 and 6, an octave apart. Their harmonic mean is 8, and 12:8 is a perfect fifth. The Pythagoreans built their theory of consonance from exactly these ratios.

Whenever a fixed quantity is shared across different rates, the harmonic mean is the honest average.

#stemantics #mathematics #harmonicmean #physics #stemeducation

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## Tags

harmonicmean, stemeducation, physics, stemantics, mathematics

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