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Alligation Rule (Mean Proportion Method)

Introduction

Alligation rule (mean proportion) एक तेज़ तरीका देता है जिससे दो अलग concentrations वाली solutions को किस ratio में मिलाना है ताकि desired concentration मिले - यह तुरंत पता लगाया जा सके। यह pattern useful है क्योंकि यह दो-component mixtures में algebra की ज़रूरत हटाता है और aptitude tests में बहुत उपयोग होता है।

यहाँ आप एक simple difference-based method और एक आसान worked example सीखेंगे।

Pattern: Alligation Rule (Mean Proportion Method)

Pattern

मुख्य विचार: दो mixtures की quantities का ratio (lower concentration : higher concentration) = (higher - mean) : (mean - lower)

Steps:
1. Lower concentration, higher concentration और required mean (target) concentration पहचानें।
2. अंतर निकालें: (higher - mean) और (mean - lower)।
3. यही दो differences lower : higher का ratio देते हैं।
4. Ratio को simplest form में बदलें और parts से actual मात्रा निकालें।

Step-by-Step Example

Question

आपके पास 10% और 30% वाली solutions हैं। 18% solution पाने के लिए इन्हें किस ratio में मिलाएँ? और अगर final solution 50 litres चाहिए, तो दोनों की कितनी मात्रा चाहिए?

Solution

  1. Step 1: Concentrations पहचानें

    Lower = 10%; Higher = 30%; Required mean = 18%.

  2. Step 2: Differences निकालें

    Lower के लिए (higher - mean) = 30 - 18 = 12

    Higher के लिए (mean - lower) = 18 - 10 = 8

  3. Step 3: Ratio बनाएँ

    Ratio (lower : higher) = 12 : 8

  4. Step 4: Ratio simplify करें

    12 : 8 = 3 : 2 (4 से divide)। यानी 10% के 3 parts और 30% के 2 parts मिलाएँ।

  5. Step 5: Final volume पर लागू करें

    Total parts = 3 + 2 = 5. 50 L के लिए एक part = 50 ÷ 5 = 10 L।

    Amount of 10% = 3 × 10 = 30 L
    Amount of 30% = 2 × 10 = 20 L

  6. Final Answer:

    Mix ratio = 3 : 2 (10% : 30%). 50 L के लिए: 30 L of 10% और 20 L of 30%.

  7. Quick Check:

    Pure substance = 30×0.10 + 20×0.30 = 3 + 6 = 9 L. Final concentration = 9 ÷ 50 = 18% ✅

Quick Variations

1. अगर ratio higher : lower चाहिए हो तो बस differences का order बदल दें।

2. जब volumes दिए हों और mean verify करना हो, तो alligation से चेक करें कि weighted average mean को मैच करता है या नहीं।

3. Cost-based mixing (value alligation) में concentrations की जगह price per unit लेते हैं - method वही रहता है।

Trick to Always Use

  • Step 1 → Lower, Mean (target), और Higher को साफ़ label करें।
  • Step 2 → (higher - mean) और (mean - lower) सही-सही निकालें; यही parts देते हैं।
  • Step 3 → Ratio simplify करें और total parts से actual quantities निकालें।

Summary

Summary

Alligation Rule (Mean Proportion Method) pattern में:

  • Mixing problem दो simple differences में बदल जाता है: (higher - mean) और (mean - lower)।
  • यही differences lower और higher की quantities का ratio देते हैं।
  • Ratio (lower : higher) = (higher - mean) : (mean - lower)।
  • Total volume पता हो तो ratio parts से actual amounts निकालें।
  • Quick check: Final concentration हमेशा दोनों concentrations के बीच होगी।

Practice

(1/5)
1. Two sugar solutions contain 10% and 40% sugar respectively. In what ratio should they be mixed to get a 20% sugar solution?
easy
A. 2 : 3
B. 1 : 2
C. 2 : 1
D. 1 : 1

Solution

  1. Step 1: Identify concentrations

    Lower = 10%, Higher = 40%, Mean = 20%.

  2. Step 2: Compute differences

    (Higher - Mean) = 40 - 20 = 20; (Mean - Lower) = 20 - 10 = 10.

  3. Step 3: Form ratio (lower : higher)

    20 : 10 = 2 : 1.

  4. Final Answer:

    Mix in ratio 2 : 1 → Option C.

  5. Quick Check:

    (2×10 + 1×40) ÷ 3 = 60 ÷ 3 = 20% ✅
Hint: Differences (higher - mean) and (mean - lower) give parts for lower and higher respectively.
Common Mistakes: Swapping which difference corresponds to which component.
2. In what ratio must 12% and 28% salt solutions be mixed to obtain a 20% solution?
easy
A. 2 : 3
B. 1 : 1
C. 3 : 2
D. 1 : 2

Solution

  1. Step 1: Identify concentrations

    Lower = 12%, Higher = 28%, Mean = 20%.

  2. Step 2: Compute differences

    (Higher - Mean) = 28 - 20 = 8; (Mean - Lower) = 20 - 12 = 8.

  3. Step 3: Form ratio (lower : higher)

    8 : 8 = 1 : 1.

  4. Final Answer:

    Mix in ratio 1 : 1 → Option B.

  5. Quick Check:

    Equal parts give (12% + 28%) ÷ 2 = 20% ✅
Hint: If differences are equal, ratio is 1 : 1 (equal parts).
Common Mistakes: Reversing difference order or failing to simplify ratios.
3. Two milk mixtures have 25% and 60% milk. In what ratio should they be mixed to get 40% milk?
easy
A. 4 : 3
B. 2 : 1
C. 3 : 2
D. 1 : 3

Solution

  1. Step 1: Identify concentrations

    Lower = 25%, Higher = 60%, Mean = 40%.

  2. Step 2: Compute differences

    (Higher - Mean) = 60 - 40 = 20; (Mean - Lower) = 40 - 25 = 15.

  3. Step 3: Form ratio (lower : higher)

    20 : 15 = 4 : 3.

  4. Final Answer:

    Mix in ratio 4 : 3 → Option A.

  5. Quick Check:

    (4×25 + 3×60) ÷ 7 = (100 + 180) ÷ 7 = 280 ÷ 7 = 40% ✅
Hint: Larger difference corresponds to the smaller concentration’s part.
Common Mistakes: Using simple average when volumes differ.
4. Two solutions of 10% and 90% are to be mixed to get a 40% solution. In what ratio should they be mixed (lower : higher)?
medium
A. 3 : 5
B. 2 : 5
C. 3 : 4
D. 5 : 3

Solution

  1. Step 1: Identify concentrations

    Lower = 10%, Higher = 90%, Mean = 40%.

  2. Step 2: Compute differences

    (Higher - Mean) = 90 - 40 = 50; (Mean - Lower) = 40 - 10 = 30.

  3. Step 3: Form ratio (lower : higher)

    50 : 30 = 5 : 3.

  4. Final Answer:

    Mix in ratio 5 : 3 → Option D.

  5. Quick Check:

    (5×10 + 3×90) ÷ 8 = (50 + 270) ÷ 8 = 320 ÷ 8 = 40% ✅
Hint: Set (higher - mean) : (mean - lower) and simplify to lowest whole numbers.
Common Mistakes: Forgetting to simplify the ratio or reversing numerator/denominator.
5. Two commodities cost ₹100/kg and ₹250/kg. In what ratio should they be mixed to get a mixture worth ₹160/kg?
medium
A. 3 : 2
B. 2 : 3
C. 1 : 1
D. 5 : 2

Solution

  1. Step 1: Identify values

    Lower = ₹100, Higher = ₹250, Mean = ₹160.

  2. Step 2: Compute differences

    (Higher - Mean) = 250 - 160 = 90; (Mean - Lower) = 160 - 100 = 60.

  3. Step 3: Form ratio (lower : higher)

    90 : 60 = 3 : 2.

  4. Final Answer:

    Mix in ratio 3 : 2 → Option A.

  5. Quick Check:

    (3×100 + 2×250) ÷ 5 = (300 + 500) ÷ 5 = 800 ÷ 5 = 160 ✅
Hint: Replace concentrations with prices and apply the same difference rule.
Common Mistakes: Confusing cheaper : costlier order when writing the ratio.

Mock Test

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