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Average Speed (Equal Distances or Times)

Introduction

Average speed problems देखने में आसान लगते हैं, लेकिन अक्सर learners दो common cases को mix कर देते हैं - जब vehicle equal distances अलग speeds पर तय करता है, और जब वह equal times अलग speeds पर चलता है।

यह pattern आपको सही formula चुनना सिखाता है ताकि आप real average speed जल्दी और accurately निकाल सकें।

Pattern: Average Speed (Equal Distances or Times)

Pattern

Key idea: Average speed = Total distance ÷ Total time. Common cases के लिए shortcuts:

  • Equal distances (दो speeds a और b): average speed = (2 × a × b) ÷ (a + b) (दो segments के लिए harmonic mean)।
  • Equal times (दो speeds a और b): average speed = (a + b) ÷ 2 (simple arithmetic mean)।

दो से ज़्यादा speeds हों तो: - Equal distances → total distance ÷ total time (generalized harmonic concept)। - Equal times → speeds का arithmetic mean।

Step-by-Step Example

Question

एक car 40 km/h की speed से 60 km चलती है और फिर 60 km/h की speed से अगला 60 km। पूरे trip की average speed क्या होगी?

Solution

  1. Step 1: Case पहचानें

    Distances equal हैं (60 km और 60 km) लेकिन speeds अलग हैं → equal-distance rule use होगा.

  2. Step 2: Harmonic shortcut apply करें

    Formula: Average = (2ab) ÷ (a + b), जहाँ a = 40, b = 60.

  3. Step 3: Calculate करें

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

  4. Alternative (full method)

    Total distance = 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:

    दोनों methods से 48 km/h आता है → correct ✔️

Quick Variations

1. Unequal distances & unequal times: हमेशा सीधे Total distance ÷ Total time use करें, कोई shortcut नहीं।

2. More than two segments: Equal times → arithmetic mean; equal distances → total distance ÷ total time।

3. Mixed units: Speeds और time को एक ही unit में convert करके फिर apply करें।

Trick to Always Use

  • Step 1: सबसे पहले पूछें: “Distances equal हैं या times equal?” - इसी से shortcut decide होता है।
  • Step 2: Equal distances → harmonic mean: (2ab)/(a+b).
  • Step 3: Equal times → arithmetic mean: (a+b)/2.
  • Step 4: Doubt हो तो हमेशा base method: Total distance ÷ Total time (100% correct)।

Summary

Summary

Key takeaways:

  • Average speed = Total distance ÷ Total time (master formula)।
  • Shortcuts: Equal distances → harmonic mean; Equal times → arithmetic mean.
  • Units हमेशा match करें और doubt होने पर full method use करें।
  • Quick check: total distance और total time से verify करें।

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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