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Average Speed

Introduction

In aptitude tests, you’ll often face problems involving average speed. Unlike normal averages, average speed is not simply the average of the two speeds.

Instead, it follows a special formula when the same distance is traveled at two different speeds.

Pattern: Average Speed

Pattern

If a person covers the same distance at two speeds, say x and y,

Average Speed = (2xy) ÷ (x + y)

This formula comes from the idea that average speed is Total Distance ÷ Total Time.

Step-by-Step Example

Question

A person travels from town A to town B at 40 km/h and returns at 60 km/h. What is the average speed for the whole trip?

Options:

  • A. 45 km/h
  • B. 48 km/h
  • C. 50 km/h
  • D. 52 km/h

Solution

  1. Step 1: Identify speeds

    Speeds given = 40 km/h and 60 km/h.

  2. Step 2: Apply correct formula

    Since the distance is the same, use:
    Average Speed = (2 × 40 × 60) ÷ (40 + 60)

  3. Step 3: Compute the result

    = 4800 ÷ 100 = 48 km/h

  4. Final Answer:

    48 km/h → Option B

  5. Quick Check:

    Total distance = 2d
    Total time = d/40 + d/60 = (5d/120) = d/24
    Avg speed = 2d ÷ (d/24) = 48 km/h → Correct ✅

Quick Variations

  • 1. If the distances are not equal, use Average Speed = Total Distance ÷ Total Time.
  • 2. For three equal distances with speeds x, y, z → Average Speed = (3xyz) ÷ (xy + yz + zx).

Trick to Always Use

  • Same distance: Use (2xy) ÷ (x + y).
  • Different distance: Use Total Distance ÷ Total Time.
  • 3 equal distances: Use the 3-term formula directly.

Summary

Summary

The Average Speed pattern is very common in aptitude exams. Always remember:

  • Equal distance (2 speeds): (2xy) ÷ (x + y)
  • Equal distance (3 speeds): (3xyz) ÷ (xy + yz + zx)
  • Unequal distance: Total Distance ÷ Total Time

Memorize these formulas to save time in exams.

Practice

(1/5)
1. A car travels 60 km at 30 km/h and returns the same distance at 90 km/h. Find the average speed.
easy
A. 45 km/h
B. 50 km/h
C. 55 km/h
D. 60 km/h

Solution

  1. Step 1: Identify speeds

    Speeds = 30 km/h and 90 km/h.

  2. Step 2: Apply formula

    Average Speed = (2xy) ÷ (x + y).

  3. Step 3: Compute

    = (2 × 30 × 90) ÷ (30 + 90) = 5400 ÷ 120 = 45 km/h.

  4. Final Answer:

    45 km/h → Option A

  5. Quick Check:

    Total distance = 120, total time = 2 + 2/3 ≈ 2.67 → 120 ÷ 2.67 ≈ 45 ✅
Hint: Use (2xy)/(x+y) directly for equal distance.
Common Mistakes: Taking simple average (30+90)/2 instead of harmonic mean.
2. A person goes to work at 40 km/h and returns at 60 km/h. What is the average speed for the trip?
easy
A. 45 km/h
B. 48 km/h
C. 50 km/h
D. 55 km/h

Solution

  1. Step 1: Identify speeds

    Speeds = 40 and 60 km/h.

  2. Step 2: Apply formula

    Average Speed = (2 × 40 × 60) ÷ (40 + 60).

  3. Step 3: Compute

    = 4800 ÷ 100 = 48 km/h.

  4. Final Answer:

    48 km/h → Option B

  5. Quick Check:

    Assume distance = 60 km each way. Time = 60/40 + 60/60 = 1.5 + 1 = 2.5 h. Average = 120 ÷ 2.5 = 48 km/h ✅
Hint: Multiply then double-check with distance/time.
Common Mistakes: Averaging 40 and 60 directly to 50.
3. A train covers equal distances at speeds of 20 km/h and 30 km/h. Find its average speed.
easy
A. 22 km/h
B. 24 km/h
C. 25 km/h
D. 26 km/h

Solution

  1. Step 1: Identify speeds

    Speeds = 20 and 30.

  2. Step 2: Apply formula

    Average Speed = (2 × 20 × 30) ÷ (20 + 30).

  3. Step 3: Compute

    = 1200 ÷ 50 = 24 km/h.

  4. Final Answer:

    24 km/h → Option B

  5. Quick Check:

    Assume distance = 30 km each way. Time = 30/20 + 30/30 = 1.5 + 1 = 2.5 h. Average = 60 ÷ 2.5 = 24 km/h ✅
Hint: For equal distances, use (2xy)/(x+y).
Common Mistakes: Using (20+30)/2 = 25 wrongly.
4. A car travels equal distances at 50 km/h and 75 km/h. Find the average speed.
medium
A. 60 km/h
B. 61 km/h
C. 62 km/h
D. 63 km/h

Solution

  1. Step 1: Identify speeds

    Speeds = 50 and 75.

  2. Step 2: Apply formula

    Average Speed = (2 × 50 × 75) ÷ (50 + 75).

  3. Step 3: Compute

    = 7500 ÷ 125 = 60 km/h.

  4. Final Answer:

    60 km/h → Option A

  5. Quick Check:

    Assume distance = 75 km each way. Time = 75/50 + 75/75 = 1.5 + 1 = 2.5 h. Average = 150 ÷ 2.5 = 60 km/h ✅
Hint: Always verify using distance/time.
Common Mistakes: Choosing 62 or 63 due to wrong calculation.
5. A person covers equal distances at 12 km/h, 15 km/h, and 20 km/h. Find the average speed.
medium
A. 14 km/h
B. 15 km/h
C. 16 km/h
D. 17 km/h

Solution

  1. Step 1: Apply correct 3-speed formula

    Average speed (equal distance) = (3xyz) ÷ (xy + yz + zx).

  2. Step 2: Substitute values

    =(3 × 12 × 15 × 20) ÷ (12×15 + 15×20 + 20×12).

  3. Step 3: Compute numerator/denominator

    = 10800 ÷ (180 + 300 + 240).

  4. Step 4: Final calculation

    = 10800 ÷ 720 = 15 km/h.

  5. Final Answer:

    15 km/h → Option B

  6. Quick Check:

    Formula and substitution validated → correct result ✅
Hint: For 3 speeds (equal distances), use (3xyz)/(xy+yz+zx).
Common Mistakes: Using the 2-speed formula instead of 3-speed formula.

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