Out at 30 mph, back at 60 — average speed?

Core

out 30 mph ⟷ back 60 mph

Press play — the board writes itself.

0:00 / 1:33

Transcript

You drive somewhere at 30 miles an hour, and you drive back at 60. Average speed for the whole trip? If your brain said 45… you're with most of the planet. And you're about to see why it's wrong.

[The round trip: an arrow out at 30 mph taking two ticked hours, and an arrow back at 60 mph taking one.]

[60 mph]

$45 ?$

45 is just 30-plus-60 over two — averaging the speeds like they're two test scores. But a speed isn't a score. A speed only counts for as long as you LIVE at it… and you lived at 30 a lot longer.

$\frac{30 + 60}{2} = 45 ?$

Watch. Nobody gave us a distance, so pick a friendly one — say 60 miles each way. 60 miles at 30 miles an hour: two full hours of crawling. 60 miles back at 60? One hour, done.

[2 h]

[1 h]

So there's the whole trip: two hours plus one hour — three hours on the road.

$\frac{60}{30} + \frac{60}{60} = 2 h + 1 h = 3 h$

And average speed was never a vibe — it's total distance over total time. 120 miles… three hours… 40 miles an hour.

$\frac{120 mi}{3 h} = 40$

Not 45 — 40. The slow leg drags the average toward itself, because you spent TWICE as long living at 30 as you did at 60.

That's the transferable move: to average rates — speeds, prices, points per game — never average the rates. Go back to the totals. Total distance over total time… and it will always lean toward wherever you spent more time.

Why the answer is 40, not 45 · 1:34

Almost everyone reaches for $\frac{30 + 60}{2} = 45$ — and the road has other plans. A speed only counts for as long as you drive at it, and the slow leg lasts twice as long as the fast one.

No distance is given, so pick a friendly one: say the trip is $60$ miles each way. Out at $30$ mph takes $\frac{60}{30} = 2$ hours; back at $60$ mph takes $\frac{60}{60} = 1$ hour. Now use the one definition that never lies:

$\text{average speed} = \frac{\text{total distance}}{\text{total time}} = \frac{120 \text{ mi}}{3 \text{ h}} = 40 \text{ mph}$

Any distance gives the same answer — try $30$ miles each way and watch the $40$ reappear.

What this hides

The word “average” quietly promises that both ingredients count equally — and here they don’t. You lived at $30$ mph for two of the three hours, so the average has to lean toward $30$ . Averaging the two speeds ignores those weights entirely; $45$ would only be true if you spent equal time at each speed, and equal distances force unequal times.

The transferable idea: a rate is a fraction, and averages of rates come from totals — total distance over total time, total cost over total pounds, total points over total games. (If you like formulas, the equal-distance shortcut is the harmonic mean, $\frac{2ab}{a+b} = \frac{2 \cdot 30 \cdot 60}{90} = 40$ — but total-over-total is the version that always works.)