Average Speed
Average speed is never the average of the speeds — it is total distance divided by total time. The most-tested CAT case is equal distances at two different speeds: there the average speed is the harmonic mean 2xy/(x + y), not (x + y)/2. Drive 60 km at 30 km/h and 60 km at 60 km/h and the average is 2×30×60/90 = 40 km/h, which is below the simple mean of 45 because you spend more time at the slower speed. The simple mean (x + y)/2 applies only when equal times are spent at each speed. The safe universal method when you are unsure is to assume a convenient total distance (often the LCM of the speeds), compute the time for each leg, then divide total distance by total time. For three equal legs the harmonic mean extends to 3xyz/(xy + yz + zx).
✅ Solved examples
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📝 Topic test — 8 questions
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Formula Reference Sheet
Core average relations
| Arithmetic mean | Average = (Sum of observations) / (Number of observations) |
|---|---|
| Sum from average | Sum = Average × Count |
| Average of first n natural numbers | (n + 1) / 2 |
| Average of an arithmetic progression | (First term + Last term) / 2 |
| Deviation (shift) method | Average = Assumed mean + (Sum of deviations) / Count |
Weighted, speed & replacement tools
| Weighted average | (w₁a₁ + w₂a₂ + …) / (w₁ + w₂ + …) |
|---|---|
| Alligation (ratio of weights) | w₁ : w₂ = (A₂ − Avg) : (Avg − A₁) |
| Average speed (whole journey) | Total distance / Total time |
| Equal-distance two speeds | 2xy / (x + y) (harmonic mean) |
| Change in average on replacement | New value = Old value ± (Change in average × Count) |