Averages

Master the concepts of averages with our comprehensive guide. Learn about arithmetic mean, weighted averages, and complex problem-solving techniques.

Table of Contents

Simplified Quantitative Formulas: Averages

  • Average (Arithmetic Mean): Average = (Sum of values) / (Number of values)
  • Weighted Average: Weighted Average = (Σ value × weight) / (Σ weights)
  • Combined Average: If two groups have averages A₁ (n₁ items) and A₂ (n₂ items), combined average = (n₁A₁ + n₂A₂)/(n₁ + n₂)
  • Effect of Adding/Removing: Adding a value x to a group changes the average by (x – old average)/n+1. Removing a value changes it by (old average – removed value)/n-1.
  • Average Speed: If a distance is covered at different speeds, average speed = total distance / total time. For equal distances at speeds S₁ and S₂, average speed = 2S₁S₂/(S₁+S₂)
  • Mean, Median, Mode: Mean = average, Median = middle value, Mode = most frequent value.
  • Variable Definitions: n = number of values, Σ = sum, A = average, w = weight, S = speed, d = distance, t = time

What do these mean? (Super Simple Explanations & Examples)

  • Average: 5, 7, 9: Average = (5+7+9)/3 = 21/3 = 7.
  • Weighted Average: 2 tests: 80 marks (weight 2), 90 marks (weight 3): (80×2 + 90×3)/(2+3) = (160+270)/5 = 86.
  • Combined Average: Group 1: 10 people, avg 60. Group 2: 20 people, avg 70. Combined = (10×60+20×70)/30 = (600+1400)/30 = 66.67
  • Effect of Adding: Old avg 10, add 20 to group of 4: New avg = (10×4+20)/5 = 12.
  • Average Speed: 60 km at 30 km/h, 60 km at 60 km/h: Avg speed = 2×30×60/(30+60) = 40 km/h.
  • Mean, Median, Mode: 2, 3, 3, 5, 7: Mean = 4, Median = 3, Mode = 3.
  • Variable Definitions: n = number of values, Σ = sum, A = average, w = weight, S = speed, d = distance, t = time

1. Basic Concepts

(a) Understanding Averages

Understanding the basic concepts of averages and their properties.

Key Terms:

  • Average: Sum of all values divided by number of values
  • Arithmetic Mean: Most common type of average
  • Weighted Average: Average considering weights of values
  • Geometric Mean: nth root of product of n values
  • Harmonic Mean: Reciprocal of average of reciprocals

Important Points:

  • Average = Sum of values/Number of values
  • Sum of values = Average × Number of values
  • If all values are increased by x, average increases by x
  • If all values are multiplied by x, average is multiplied by x

Example 1: Basic Average

Q1.

Find the average of 15, 20, 25, 30, 35

Solution:

Sum of numbers = 15 + 20 + 25 + 30 + 35 = 125

Number of terms = 5

Average = 125/5 = 25

Q2.

The average of 5 numbers is 20. If one number is removed, the average becomes 18. Find the removed number.

Solution:

Sum of 5 numbers = 20 × 5 = 100

Sum of 4 numbers = 18 × 4 = 72

Removed number = 100 - 72 = 28

Q3.

The average age of a class of 30 students is 15 years. If the teacher's age is included, the average becomes 16 years. Find the teacher's age.

Solution:

Sum of students' ages = 15 × 30 = 450

Sum including teacher = 16 × 31 = 496

Teacher's age = 496 - 450 = 46 years

Q4.

The average of 8 numbers is 25. If each number is increased by 5, find the new average.

Solution:

When each number is increased by 5, the average also increases by 5

New average = 25 + 5 = 30

2. Arithmetic Mean

(a) Arithmetic Mean Problems

Solving problems involving arithmetic mean.

Formulas:

  • Arithmetic Mean = Sum of values/Number of values
  • Sum of values = Mean × Number of values
  • Number of values = Sum of values/Mean

Example 2: Arithmetic Mean

Q1.

The average of 6 numbers is 30. If one number is excluded, the average becomes 28. Find the excluded number.

Solution:

Sum of 6 numbers = 30 × 6 = 180

Sum of 5 numbers = 28 × 5 = 140

Excluded number = 180 - 140 = 40

Q2.

The average of 10 numbers is 25. If each number is multiplied by 2, find the new average.

Solution:

When each number is multiplied by 2, the average is also multiplied by 2

New average = 25 × 2 = 50

Q3.

The average of 7 numbers is 20. If the average of first 4 numbers is 18 and the average of last 4 numbers is 22, find the 4th number.

Solution:

Sum of 7 numbers = 20 × 7 = 140

Sum of first 4 numbers = 18 × 4 = 72

Sum of last 4 numbers = 22 × 4 = 88

4th number = 72 + 88 - 140 = 20

Q4.

The average of 5 numbers is 15. If the average of first 3 numbers is 12 and the average of last 3 numbers is 18, find the 3rd number.

Solution:

Sum of 5 numbers = 15 × 5 = 75

Sum of first 3 numbers = 12 × 3 = 36

Sum of last 3 numbers = 18 × 3 = 54

3rd number = 36 + 54 - 75 = 15

3. Weighted Averages

(a) Weighted Average Problems

Solving problems involving weighted averages.

Formulas:

  • Weighted Average = Σ(Value × Weight)/Σ(Weight)
  • For two components:
  • Average = (a₁w₁ + a₂w₂)/(w₁ + w₂)
  • Where:
  • a₁, a₂ = Values
  • w₁, w₂ = Weights

Example 3: Weighted Average

Q1.

A student scored 80 in Math (weight 4), 75 in Science (weight 3), and 90 in English (weight 2). Find the weighted average.

Solution:

Weighted sum = (80 × 4) + (75 × 3) + (90 × 2)

Weighted sum = 320 + 225 + 180 = 725

Total weight = 4 + 3 + 2 = 9

Weighted average = 725/9 = 80.56

Q2.

In a class, 20 students scored an average of 75 marks, and 30 students scored an average of 85 marks. Find the overall average.

Solution:

Total marks of first group = 75 × 20 = 1500

Total marks of second group = 85 × 30 = 2550

Total marks = 1500 + 2550 = 4050

Total students = 20 + 30 = 50

Overall average = 4050/50 = 81

Q3.

A company has 40 employees with an average salary of ₹30,000 and 60 employees with an average salary of ₹40,000. Find the average salary of all employees.

Solution:

Total salary of first group = 30000 × 40 = ₹12,00,000

Total salary of second group = 40000 × 60 = ₹24,00,000

Total salary = ₹36,00,000

Total employees = 100

Average salary = ₹36,00,000/100 = ₹36,000

Q4.

A student's final grade is based on: Midterm (30%), Final (50%), and Projects (20%). If the student scored 85, 90, and 95 respectively, find the final grade.

Solution:

Weighted sum = (85 × 0.3) + (90 × 0.5) + (95 × 0.2)

Weighted sum = 25.5 + 45 + 19 = 89.5

Final grade = 89.5

4. Geometric Mean

(a) Geometric Mean Problems

Solving problems involving geometric mean.

Formulas:

  • Geometric Mean = nth root of (a₁ × a₂ × ... × aₙ)
  • For two numbers:
  • GM = √(a₁ × a₂)
  • For three numbers:
  • GM = ∛(a₁ × a₂ × a₃)

Example 4: Geometric Mean

Q1.

Find the geometric mean of the numbers 4 and 9.

Solution:

GM = √(4 × 9) = √36 = 6

Q2.

Find the geometric mean of 2, 4, and 8.

Solution:

GM = ∛(2 × 4 × 8) = ∛64 = 4

Q3.

The geometric mean of two numbers is 6 and their arithmetic mean is 6.5. Find the numbers.

Solution:

Let numbers be a and b

√(ab) = 6 ⇒ ab = 36

(a + b)/2 = 6.5 ⇒ a + b = 13

Solving: a = 4, b = 9

Q4.

The geometric mean of three numbers is 4 and their arithmetic mean is 4.5. If two numbers are 2 and 8, find the third number.

Solution:

Let third number be x

∛(2 × 8 × x) = 4 ⇒ 16x = 64 ⇒ x = 4

5. Harmonic Mean

(a) Harmonic Mean Problems

Solving problems involving harmonic mean.

Formulas:

  • Harmonic Mean = n/(1/a₁ + 1/a₂ + ... + 1/aₙ)
  • For two numbers:
  • HM = 2ab/(a + b)
  • For three numbers:
  • HM = 3abc/(ab + bc + ca)

Example 5: Harmonic Mean

Q1.

Find the harmonic mean of 2 and 3.

Solution:

HM = 2(2)(3)/(2 + 3) = 12/5 = 2.4

Q2.

Find the harmonic mean of 1, 2, and 3.

Solution:

HM = 3(1)(2)(3)/(1×2 + 2×3 + 3×1) = 18/11 ≈ 1.636

Q3.

The harmonic mean of two numbers is 4 and their geometric mean is 4.5. Find the numbers.

Solution:

Let numbers be a and b

2ab/(a + b) = 4 ⇒ ab = 2(a + b)

√(ab) = 4.5 ⇒ ab = 20.25

Solving: a = 4.5, b = 4.5

Q4.

The harmonic mean of three numbers is 3 and their arithmetic mean is 4. If two numbers are 2 and 4, find the third number.

Solution:

Let third number be x

3(2)(4)(x)/(2×4 + 4×x + x×2) = 3

24x/(8 + 4x + 2x) = 3

24x = 24 + 18x

6x = 24

x = 4

6. Advanced Concepts

(a) Important Theorems

Key Theorems:

  1. For any set of positive numbers:
    AM ≥ GM ≥ HM
  2. For two numbers a and b:
    AM = (a + b)/2

    GM = √(ab)

    HM = 2ab/(a + b)
  3. For n numbers:
    AM = Σaᵢ/n

    GM = nth root of Πaᵢ

    HM = n/Σ(1/aᵢ)

Example 6: Complex Problems

Q1.

The arithmetic mean of 5 numbers is 20. If each number is increased by 2, find the new arithmetic mean.

Solution:

When each number is increased by 2, the mean also increases by 2

New mean = 20 + 2 = 22

Q2.

The geometric mean of 4 numbers is 8. If each number is multiplied by 3, find the new geometric mean.

Solution:

When each number is multiplied by 3, the geometric mean is also multiplied by 3

New geometric mean = 8 × 3 = 24

Q3.

The harmonic mean of 3 numbers is 6. If each number is divided by 2, find the new harmonic mean.

Solution:

When each number is divided by 2, the harmonic mean is also divided by 2

New harmonic mean = 6/2 = 3

Q4.

The arithmetic mean of 6 numbers is 15. If the average of first 3 numbers is 12 and the average of last 3 numbers is 18, find the 3rd number.

Solution:

Sum of 6 numbers = 15 × 6 = 90

Sum of first 3 numbers = 12 × 3 = 36

Sum of last 3 numbers = 18 × 3 = 54

3rd number = 36 + 54 - 90 = 0

(b) Special Cases

Special Scenarios:

  • When all numbers are equal:
    AM = GM = HM
  • When numbers are in geometric progression:
    GM = √(first × last)
  • When numbers are in arithmetic progression:
    AM = (first + last)/2

Example 7: Special Cases

Q1.

Find the arithmetic mean of numbers in AP: 2, 5, 8, 11, 14

Solution:

AM = (2 + 14)/2 = 8

Q2.

Find the geometric mean of numbers in GP: 2, 4, 8, 16

Solution:

GM = √(2 × 16) = √32 = 4√2

Q3.

If AM = GM = HM for two numbers, find the numbers.

Solution:

When AM = GM = HM, the numbers must be equal

Let the numbers be x and x

AM = (x + x)/2 = x

GM = √(x × x) = x

HM = 2x²/(x + x) = x

Therefore, the numbers are equal

Q4.

The arithmetic mean of 5 numbers is 20. If the geometric mean is 18, find the harmonic mean.

Solution:

For 5 numbers, we know:

AM ≥ GM ≥ HM

20 ≥ 18 ≥ HM

Therefore, HM ≤ 18

Without more information, we cannot determine the exact value of HM

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