Time, Speed & Distance

Distance = Speed × Time. Everything else is a rearrangement — but only if the three units agree.

Conversions to memorise: km/h → m/s: × 5/18; m/s → km/h: × 18/5. 36 km/h is 10 m/s; 54 km/h is 15 m/s; 72 km/h is 20 m/s.

Four types:

1. Plain D = S × T, usually with a unit change hidden in it — the answer is wanted in minutes but the speed is per hour.
2. Average speed. total distance ÷ total time, always. For two equal distances it works out to 2xy/(x+y); for two equal times it is the plain mean (x+y)/2.
3. Late and early. "At x km/h he is p minutes late, at y km/h he is q minutes early." The time gap is p + q minutes — they add, because one is after the deadline and the other before. Then d/x − d/y = (p+q)/60.
4. Same time, different distance. "Had he walked at y instead of x, he would have gone k km more." Since the time is equal, d/x = (d+k)/y.

The single most common mistake is averaging the two speeds in type 2. On equal distances, more time is spent at the slower speed, so the true average always sits below the arithmetic mean. If your answer is exactly halfway between the two speeds, check whether the equal quantity was distance or time.

D = S × T · km/h → m/s: × 5/18 · average speed = total distance/total time · equal distances → 2xy/(x+y) · late p + early q = time gap

A man covers 30 km at 10 km/h and the next 30 km at 15 km/h. Find his average speed. → Time = 30/10 + 30/15 = 3 + 2 = 5 hours for 60 km, so the average is 60 ÷ 5 = 12 km/h. Sanity check: the equal-distance shortcut gives 2 × 10 × 15/(10 + 15) = 300/25 = 12, and note that 12 is below the plain mean of 12.5 — as it must be, because he spent 3 of the 5 hours at the slower speed.