Arithmetic Mean (AM) & Harmonic Mean (HM) Use
1. Arithmetic Mean (AM): Definition:- Arithmetic mean is the simple average of given numbers.
Formula: Arithmetic Mean = a+b / 2
When to Use: When time spent is equal at different speeds.
Ex: A car runs at 40 km/h for 1 hour and 60 km/h for 1 hour.
AM = 40 + 60 / 2 = 50 km/h
2. Harmonic Mean (HM): Definition:- Harmonic mean is the average of rates, used when distance is equal.
Formula: Harmonic Mean = 2ab / a+b
When to Use: When equal distances are covered at different speeds.
Ex: A bus goes at 60 km/h and returns at 40 km/h over the same distance.
HM = 2⋅60⋅40 / 60+40 = 48 km/h
Arithmetic Mean (AM)
- Used when time spent is equal.
- Applies to simple averaging.
- Formula: (a + b) ÷ 2
- Example: Same time at 40 km/h and 60 km/h → AM = 50 km/h.
Harmonic Mean (HM)
- Used when distance covered is equal.
- Applies to average speed problems.
- Formula: 2ab ÷ (a + b)
- Example: Same distance at 60 km/h and 40 km/h → HM = 48 km/h.
Rule: Equal Time → Arithmetic Mean // Equal Distance → Harmonic Mean
Memory Trick: Same time → Arithmetic Mean // Same distance → Harmonic Mean
Q. A bus travels 90 km/h on the onward journey and 60 km/h on the return journey. What is the average speed of the bus for the entire trip?
Options: (a) 70 km/h (b) 72 km/h (c) 75 km/h (d) 78 km/h
Soln
When a vehicle covers equal distances at different speeds:
Average speed = 2⋅V1⋅V2 / V1+V2
Where V1 = 90 ,V2 = 60 km/h
Substitute the values
Average speed = 2⋅90⋅60 / 90+60 = 10800 /150 = 72 km/h
Ans: (b) 72 km/h
Q. A car travels 40 km/h while going and 60 km/h while returning over the same distance. What is the average speed ?
Options: (a) 48 km/h (b) 50 km/h (c) 52 km/h (d) 55 km/h
Soln:
Average speed = 2⋅40⋅60 /40+60 = 4800/100 = 48 km/h
Ans: (a) 48 km/h
Trick: Average speed is always less than the arithmetic mean (which is 50), so 48 fits.
Q: A bike covers the same distance at 30 km/h and 45 km/h. Find the average speed.
Options: (a) 36 km/h (b) 37.5 km/h (c) 40 km/h (d) 42 km/h
Soln:
Average speed = 2⋅30⋅45 / 30+45 = 2700/75 = 36 km/h
Ans: (a) 36 km/h
Trick: Multiply both speeds ×2 and divide by their sum — no distance needed.
Memory Tip: “Equal distance → harmonic mean; Equal time → arithmetic mean.”
Ex 1: Equal Distance → Harmonic Mean
Q. A car travels 60 km/h on the onward journey and 40 km/h on the return journey over the same distance. What is the average speed?
Options: (a) 45 km/h (b) 48 km/h (c) 50 km/h (d) 52 km/h
Soln:
Average speed = 2⋅60⋅40 / 60+40 = 4800 / 100=48 km/h
Ans: (b) 48 km/h
Key Note: Average is closer to the slower speed (40) → use harmonic mean.
Ex 2: Equal Time → Arithmetic Mean
Q. A bike runs at 40 km/h for 1 hour and then at 60 km/h for 1 hour. What is the average speed ?
Options: (a) 45 km/h (b) 48 km/h (c) 50 km/h (d) 52 km/h
Soln :
Average speed = 40+60 / 2 = 50 km/h
Ans: (c) 50 km/h
Key Note: Time is the same → use arithmetic mean.
Rule: Equal distance → Harmonic mean | Equal time → Arithmetic mean
Q. A man goes from home to office at a speed of 80 km/h and returns from office to home at a speed of x km/h. If the average speed for the whole journey is 96 km/h, find the value of x.
Options: (a) 100 km/h (b) 110 km/h (c) 120 km/h (d) 150 km/h
Soln:
Distance is the same in both directions, we use the harmonic mean formula:
Average speed = 2ab / a+b
Here,
a = 80km/h,
b = x km/h,
Average speed = 96 km/h
96 = 2*80*X /80 + x
= 96(80 + x) = 160x
= 7680 + 96x = 160x
= 7680 = 64x
= 240 = 2X
= X = 120
Quick Exam Shortcut:
- When average speed is greater than one speed, the other speed must be much higher.
- Since average (96) is greater than 80, return speed must be above 96.
- Here options, 120 km/h fits.