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

Introduction

Arithmetic Mean (A.M.), Geometric Mean (G.M.), और Harmonic Mean (H.M.) के बीच का संबंध numerical reasoning का बहुत महत्वपूर्ण हिस्सा है। यह additive, multiplicative और rate-based averages को compare करने में मदद करता है और कई algebra, statistics और ratio वाले questions में तेजी से answer verify करने में काम आता है।

यह pattern इसलिए उपयोगी है क्योंकि इससे हमेशा यह पता चलता है कि कौन-सा mean सबसे बड़ा/छोटा होगा - जो कई competitive exam questions को छोटा कर देता है।

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

Pattern

किसी भी positive numbers के set के लिए हमेशा: A.M. ≥ G.M. ≥ H.M. होता है - equality तभी होगी जब सभी numbers equal हों।

अगर numbers x₁, x₂, …, xₙ (सब > 0) हों, तो:

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

Main inequality idea (two-number proof): Positive a, b के लिए → (a + b)/2 ≥ √(ab) (A.M.-G.M.), और reciprocals पर यही apply करने से G.M.-H.M. वाला relation मिलता है।

Step-by-Step Example

Question

Numbers 4 और 9 के लिए A.M., G.M., H.M. निकालें और verify करें कि A.M. ≥ G.M. ≥ H.M.

Solution

  1. Step 1: A.M. निकालें

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

  2. Step 2: G.M. निकालें

    G.M. = √(4 × 9) = √36 = 6

  3. Step 3: H.M. निकालें

    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:

    Numbers equal नहीं हैं → इसलिए inequality strict है (equality नहीं होगी) ✅

Quick Variations

1. n numbers के लिए भी यही formulas directly apply होते हैं - inequality हमेशा hold करती है।

2. Weighted means में भी (non-negative weights) weighted A.M. ≥ weighted G.M. रहता है।

3. Reciprocal trick: H.M. = 1 / (A.M. of reciprocals) → इससे G.M. और H.M. compare करना आसान हो जाता है।

Trick to Always Use

  • Step 1: दो numbers की quick check के लिए (a + b)/2 ≥ √(ab) का use करें।
  • Step 2: G.M. और H.M. compare करने के लिए reciprocals का A.M.-G.M. rule लगाएँ।

Summary

Summary

मुख्य बातें:

  • A.M. ≥ G.M. ≥ H.M.: किसी भी positive set के लिए हमेशा सही।
  • A.M. additive averages के लिए, G.M. growth/ratio-based changes के लिए, और H.M. rates (जैसे speed) के लिए सबसे useful होता है।
  • Quick verification → दो means निकालकर उनकी ordering check कर लें।

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

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