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Average Speed (Equal Distances or Times)

Introduction

Average speed problems often look simple but trap learners when they confuse the two common cases: when the vehicle covers equal distances at different speeds, and when it moves for equal times at different speeds.

This pattern teaches clear rules to pick the right formula so you compute the true average speed quickly and accurately.

Pattern: Average Speed (Equal Distances or Times)

Pattern

Key idea: Average speed = Total distance ÷ Total time. Use shortcuts for common cases:

  • Equal distances (two speeds a and b): average speed = (2 × a × b) ÷ (a + b) (harmonic-mean formula for two segments).
  • Equal times (two speeds a and b): average speed = (a + b) ÷ 2 (simple arithmetic mean).

For more than two speeds: - Equal distances → use generalized harmonic mean (total distance ÷ total time). - Equal times → arithmetic mean of speeds.

Step-by-Step Example

Question

A car travels 60 km at 40 km/h and then another 60 km at 60 km/h. What is the average speed for the entire trip?

Solution

  1. Step 1: Identify the case

    Distances are equal (60 km and 60 km) but speeds are different → use the equal-distances rule.

  2. Step 2: Apply the harmonic shortcut (or compute total time)

    Shortcut formula for two equal distances: Average speed = (2 × a × b) ÷ (a + b), where a = 40, b = 60.

  3. Step 3: Substitute and calculate (shortcut)

    Average = (2 × 40 × 60) ÷ (40 + 60) = (4800) ÷ 100 = 48 km/h.

  4. Alternative computation (full method)

    Total distance = 60 + 60 = 120 km. Time1 = 60 ÷ 40 = 1.5 h; Time2 = 60 ÷ 60 = 1 h. Total time = 2.5 h. Average speed = 120 ÷ 2.5 = 48 km/h.

  5. Final Answer:

    Average speed = 48 km/h.

  6. Quick Check:

    Using either method yields 48 km/h → consistent ✅

Quick Variations

1. Unequal distances & unequal times: Always use Total distance ÷ Total time (no shortcut).

2. More than two segments: For equal times → arithmetic mean of all speeds; for equal distances → use total distance ÷ total time (or harmonic mean extension).

3. Mixed units: Convert speeds/times to consistent units (e.g., km/h and hours) before using shortcuts.

Trick to Always Use

  • Step 1: Ask: "Are distances equal or times equal?" - this decides the shortcut.
  • Step 2: If equal distances → use harmonic formula: (2ab)/(a+b) for two speeds.
  • Step 3: If equal times → use arithmetic mean: (a+b)/2 for two speeds.
  • Step 4: If unsure, compute Total distance and Total time and do Distance ÷ Time (always correct).

Summary

Summary

Key takeaways:

  • Average speed = Total distance ÷ Total time (master formula).
  • Shortcuts: Equal distances → harmonic mean; Equal times → arithmetic mean.
  • Always check units and use the full method when in doubt.
  • Quick check: recompute total distance and total time to verify your result.

Practice

(1/5)
1. A car covers 60 km at 40 km/h and the next 60 km at 60 km/h. Find its average speed.
easy
A. 45 km/h
B. 48 km/h
C. 46 km/h
D. 50 km/h

Solution

  1. Step 1: Identify the Case

    Distances are equal, so use the equal-distance (harmonic mean) formula.

  2. Step 2: Apply the Formula

    Average speed = (2 × a × b) ÷ (a + b).

  3. Step 3: Substitute and Calculate

    Average = (2 × 40 × 60) ÷ (40 + 60) = 4800 ÷ 100 = 48 km/h.

  4. Final Answer:

    Average speed = 48 km/h → Option B.

  5. Quick Check:

    Total distance = 120 km; total time = 2.5 h → 120 ÷ 2.5 = 48 ✅
Hint: For equal distances, use (2ab)/(a+b).
Common Mistakes: Taking simple average instead of harmonic mean.
2. A train travels for 2 hours at 60 km/h and for another 2 hours at 100 km/h. Find its average speed.
easy
A. 80 km/h
B. 75 km/h
C. 85 km/h
D. 90 km/h

Solution

  1. Step 1: Identify the Case

    Times are equal → use arithmetic mean formula.

  2. Step 2: Apply the Formula

    Average speed = (a + b) ÷ 2.

  3. Step 3: Calculate

    Average = (60 + 100) ÷ 2 = 160 ÷ 2 = 80 km/h.

  4. Final Answer:

    Average speed = 80 km/h → Option A.

  5. Quick Check:

    Total distance = 60×2 + 100×2 = 320 km; total time = 4 h → 320 ÷ 4 = 80 ✅
Hint: For equal times, take the simple average of speeds.
Common Mistakes: Applying harmonic formula instead of arithmetic mean.
3. A cyclist goes 30 km at 20 km/h and returns the same distance at 30 km/h. Find his average speed.
easy
A. 22 km/h
B. 23 km/h
C. 24 km/h
D. 25 km/h

Solution

  1. Step 1: Identify the Case

    Equal distances (30 km each way) → use harmonic mean formula.

  2. Step 2: Apply Formula

    Average speed = (2 × a × b) ÷ (a + b).

  3. Step 3: Substitute and Calculate

    Average = (2 × 20 × 30) ÷ (20 + 30) = 1200 ÷ 50 = 24 km/h.

  4. Final Answer:

    Average speed = 24 km/h → Option C.

  5. Quick Check:

    Total distance = 60 km; total time = (30÷20)+(30÷30)=1.5 + 1 = 2.5 h → 60 ÷ 2.5 = 24 ✅
Hint: Use (2ab)/(a+b) for to-and-fro equal distance trips.
Common Mistakes: Taking (20+30)/2 instead of harmonic mean.
4. A bus covers equal distances at speeds of 30 km/h, 40 km/h, and 60 km/h. Find its average speed.
medium
A. 40 km/h
B. 38 km/h
C. 36 km/h
D. 42 km/h

Solution

  1. Step 1: Identify the Case

    Equal distances → use harmonic mean formula extended for three speeds.

  2. Step 2: Formula

    Average speed = n ÷ (Σ(1/a)) where n = 3.

  3. Step 3: Calculate

    Average = 3 ÷ ((1/30)+(1/40)+(1/60)) = 3 ÷ ((4+3+2)/120) = 3 ÷ (9/120) = 3 × (120/9) = 40 km/h.

  4. Final Answer:

    Average speed = 40 km/h → Option A.

  5. Quick Check:

    Total distance = 180 km; total time = 4.5 h → 180 ÷ 4.5 = 40 ✅
Hint: For 3 equal distances, use 3 / (1/a + 1/b + 1/c).
Common Mistakes: Using arithmetic mean instead of harmonic mean.
5. A car travels 100 km at 50 km/h, then 100 km at 25 km/h, and then 100 km at 75 km/h. Find the average speed.
medium
A. 40 km/h
B. 42 km/h
C. 43 km/h
D. 41 km/h

Solution

  1. Step 1: Identify the Case

    Equal distances (100 km each) → harmonic mean for three speeds.

  2. Step 2: Formula

    Average speed = 3 ÷ (Σ(1/a)) = 3 ÷ ((1/50)+(1/25)+(1/75)).

  3. Step 3: Calculate

    (1/50)+(1/25)+(1/75) = 0.02 + 0.04 + 0.013333... = 0.073333... ⇒ Average = 3 ÷ 0.073333... = 40.909... km/h ≈ 41 km/h.

  4. Final Answer:

    Average speed = 41 km/h → Option D.

  5. Quick Check:

    Total distance = 300 km; total time = 100/50 + 100/25 + 100/75 = 2 + 4 + 1.333... = 7.333... h → 300 ÷ 7.333... ≈ 40.909... ✅
Hint: For 3 equal distances, use harmonic mean formula 3 / (Σ(1/speed)).
Common Mistakes: Using simple average instead of harmonic mean.

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