How to Solve Average Speed for Two-Part Journeys
Solve average speed aptitude problems for two-part journeys using the harmonic mean and arithmetic mean, with worked examples.
Expected Interview Answer
Average speed for a journey is always total distance divided by total time, never the simple average of the two speeds; when equal distances are covered at speeds a and b, the average speed simplifies to the harmonic mean, 2ab/(a+b).
The mistake most candidates make is averaging the two speeds directly, which only works when the two segments take equal time, not equal distance. For equal distances D at speeds a and b, total distance is 2D and total time is D/a + D/b, giving average speed 2D / (D/a + D/b) = 2ab/(a+b) — the harmonic mean, which is always less than or equal to the arithmetic mean (a+b)/2. When the two segments instead take equal time, the simple average (a+b)/2 is correct, because both speeds contribute equally to the total distance. Always identify whether the problem fixes distance or time before choosing a formula.
- The 2ab/(a+b) shortcut instantly solves equal-distance two-speed problems
- Recognizing equal-time cases prevents applying the harmonic mean incorrectly
- Total distance over total time is a single rule that never fails, even without the shortcut
AI Mentor Explanation
A batter scores at a strike rate of 100 for the first half of their innings by balls faced, then at strike rate 150 for the second half — if both halves used the same number of balls (equal “distance” in ball count is not it, equal time-on-strike matters here), the true combined strike rate is total runs over total balls, not the simple average of 100 and 150. When the split is by equal balls faced (equal distance) at two different scoring rates, the correct combined rate uses the harmonic-mean-style formula, 2ab/(a+b), because time spent per ball differs.
Worked example
First half
- 120km at 40 km/h
- = 3h
Second half
- 120km at 60 km/h
- = 2h
Average speed
- 240/5 = 48 km/h
- = 2ab/(a+b)
Step-by-Step Explanation
Step 1
Check what is equal
Determine if the two segments have equal distance or equal time.
Step 2
Equal distance case
Use the harmonic mean: average speed = 2ab/(a+b).
Step 3
Equal time case
Use the simple arithmetic mean: average speed = (a+b)/2.
Step 4
Fallback method
When unsure or segments are uneven, compute total distance ÷ total time directly.
What Interviewer Expects
- Recognition that average speed is never a simple average by default
- Correct application of the harmonic mean for equal-distance journeys
- Correct application of the simple mean for equal-time journeys
- Fallback to total distance ÷ total time when structure is unclear
Common Mistakes
- Always taking the simple average of two speeds regardless of the equal-distance/equal-time distinction
- Misapplying the harmonic mean formula to an equal-time scenario
- Forgetting to double-check with total distance ÷ total time
- Arithmetic errors computing 2ab/(a+b)
Best Answer (HR Friendly)
“The trap here is assuming average speed is just the average of the two speeds — that is only true if the two parts of the journey take equal time. If instead the two parts cover equal distances, which is the more common setup, the correct average speed is the harmonic mean, 2ab over a plus b, because the slower leg eats up more time for the same distance. When in doubt, I just fall back to total distance divided by total time, which always works.”
Follow-up Questions
- Why is the harmonic mean always less than or equal to the arithmetic mean here?
- How would a three-segment equal-distance journey change the formula?
- How do you handle average speed when both distance and time vary across segments?
- How does this concept extend to average rate problems in work-and-time questions?
MCQ Practice
1. A cyclist rides the first half of a distance at 15 km/h and the second half at 30 km/h. Average speed for the trip is?
Harmonic mean: 2×15×30/(15+30) = 900/45 = 20 km/h.
2. A car travels for the first half of the TIME at 50 km/h and the second half of the TIME at 70 km/h. Average speed is?
Equal time case uses the simple average: (50+70)/2 = 60 km/h.
3. Which formula applies when a journey covers equal DISTANCES at two different speeds a and b?
Equal-distance journeys use the harmonic mean, 2ab/(a+b).
Flash Cards
Average speed general rule? — Always total distance ÷ total time — never a simple average by default.
Equal-distance two-speed formula? — Harmonic mean: 2ab/(a+b).
Equal-time two-speed formula? — Simple arithmetic mean: (a+b)/2.
Why is harmonic mean ≤ arithmetic mean? — The slower speed’s longer duration for equal distance pulls the average down.