0
0

Relationship Between A.M., G.M., and H.M.

Introduction

The relationships between Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.) are fundamental in numerical reasoning. They help compare averages from different contexts (additive, multiplicative, rates) and provide inequality bounds that are widely used in problem solving.

This pattern is important because it gives a quick way to rank means and verify answers in algebra, statistics, and ratio problems.

Pattern: Relationship Between A.M., G.M., and H.M.

Pattern

For any set of positive numbers, A.M. ≥ G.M. ≥ H.M. with equality only when all numbers are equal.

Definitions for numbers x₁, x₂, …, xₙ (all > 0):

  • A.M. = (x₁ + x₂ + … + xₙ) / n
  • G.M. = (x₁ × x₂ × … × xₙ)^(1/n)
  • H.M. = n / (1/x₁ + 1/x₂ + … + 1/xₙ)

Key inequality (two-number case proof idea): For positive a,b → (a + b)/2 ≥ √(ab) (A.M.-G.M.). Then G.M.-H.M. follows by applying A.M.-G.M. to reciprocals.

Step-by-Step Example

Question

For the numbers 4 and 9, compute A.M., G.M., H.M. and verify A.M. ≥ G.M. ≥ H.M.

Solution

  1. Step 1: Compute A.M.

    A.M. = (4 + 9) / 2 = 13 / 2 = 6.5.

  2. Step 2: Compute G.M.

    G.M. = √(4 × 9) = √36 = 6.

  3. Step 3: Compute H.M.

    H.M. = 2 / (1/4 + 1/9) = 2 / (9/36 + 4/36) = 2 / (13/36) = 2 × (36/13) = 72/13 ≈ 5.5385.

  4. Step 4: Compare

    6.5 (A.M.) ≥ 6 (G.M.) ≥ 5.5385 (H.M.) → Verified.

  5. Quick Check:

    All numbers are not equal → strict inequalities hold (no equality case) ✅

Quick Variations

1. n numbers: compute using general formulas - inequality still holds.

2. Weighted means: weighted A.M. ≥ weighted G.M. under nonnegative weights (use generalized A.M.-G.M.).

3. Reciprocal trick: to compare G.M. and H.M. easily, note H.M. of xᵢ = 1 / (A.M. of reciprocals).

Trick to Always Use

  • Step 1 → Use two-number A.M.-G.M.: For quick checks with two numbers a,b, verify (a + b)/2 ≥ √(ab).
  • Step 2 → Use reciprocals: To compare G.M. and H.M., apply A.M.-G.M. to {1/xᵢ} to get 1/(A.M. of reciprocals) ≤ G.M., which rearranges to G.M. ≥ H.M.

Summary

Summary

Key takeaways:

  • A.M. ≥ G.M. ≥ H.M. for any positive set of numbers - equal only when all numbers are equal.
  • Use A.M. formula for additive averages, G.M. for multiplicative growth, and H.M. for rates/ratios (e.g., average speed).
  • To verify quickly: compute two of the means numerically and check ordering; use reciprocals to move between G.M. and H.M.

Practice

(1/5)
1. Find the Arithmetic Mean (A.M.) between 6 and 10.
easy
A. 8
B. 7
C. 9
D. 10

Solution

  1. Step 1: Apply the A.M. formula for two numbers

    A.M. = (a + b) / 2.

  2. Step 2: Substitute values

    A.M. = (6 + 10) / 2 = 16 / 2 = 8.

  3. Final Answer:

    Arithmetic Mean = 8 → Option A.

  4. Quick Check:

    8 is exactly midway between 6 and 10: 6 + 2 = 8 and 8 + 2 = 10 ✅

Hint: For two numbers, A.M. = (sum)/2 - think 'midpoint'.
Common Mistakes: Using G.M. (√(ab)) instead of A.M.
2. Find the Geometric Mean (G.M.) between 4 and 9.
easy
A. 5
B. 6
C. 7
D. 8

Solution

  1. Step 1: Use the G.M. formula for two numbers

    G.M. = √(a × b).

  2. Step 2: Substitute values

    G.M. = √(4 × 9) = √36 = 6.

  3. Final Answer:

    Geometric Mean = 6 → Option B.

  4. Quick Check:

    6² = 36 = 4×9, so geometric mean is correct ✅

Hint: For two numbers, G.M. = √(product).
Common Mistakes: Calculating (a + b)/2 instead of √(ab).
3. Find the Harmonic Mean (H.M.) between 6 and 12.
easy
A. 8
B. 9
C. 10
D. 11

Solution

  1. Step 1: Use the two-number H.M. formula

    H.M. = 2ab / (a + b).

  2. Step 2: Substitute values

    H.M. = (2 × 6 × 12) / (6 + 12) = 144 / 18 = 8.

  3. Final Answer:

    Harmonic Mean = 8 → Option A.

  4. Quick Check:

    A.M. = 9, G.M. ≈ 8.485, H.M. = 8 → ordering A.M. ≥ G.M. ≥ H.M. holds ✅

Hint: H.M. weights smaller values more: H.M. = 2ab/(a+b) for two numbers.
Common Mistakes: Using (a + b)/2 (A.M.) instead of harmonic formula.
4. If A.M. = 10 and G.M. = 8 for two positive numbers, find H.M. using the relation A.M. × H.M. = (G.M.)².
medium
A. 6.2
B. 6.5
C. 32/5
D. 6.8

Solution

  1. Step 1: Use the identity A.M. × H.M. = (G.M.)²

    Substitute given values: 10 × H.M. = 8².

  2. Step 2: Solve for H.M.

    10 × H.M. = 64 ⇒ H.M. = 64 / 10 = 32/5 (which is 6.4).

  3. Final Answer:

    Harmonic Mean = 32/5 → Option C.

  4. Quick Check:

    10 × (32/5) = 64 = 8², identity holds ✅

Hint: Use A.M.×H.M. = (G.M.)² to find the missing mean directly.
Common Mistakes: Dividing by A.M. incorrectly or rounding before final check.
5. If the A.M. and H.M. of two positive numbers are 16 and 9 respectively, find their G.M.
medium
A. 10
B. 11
C. 13
D. 12

Solution

  1. Step 1: Use relation A.M. × H.M. = (G.M.)²

    Substitute values: 16 × 9 = (G.M.)².

  2. Step 2: Compute and take square root

    (G.M.)² = 144 ⇒ G.M. = √144 = 12.

  3. Final Answer:

    Geometric Mean = 12 → Option D.

  4. Quick Check:

    16 × 9 = 144 and 12² = 144, so relation holds ✅

Hint: Compute G.M. as √(A.M. × H.M.).
Common Mistakes: Taking square root of wrong product or misreading A.M./H.M. values.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes