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Profit-Based Mixture (Value Alligation)

Introduction

कई mixture problems में items अलग-अलग cost पर खरीदे जाते हैं या अलग selling prices पर बेचे जाते हैं और फिर उन्हें मिलाया जाता है। Value Alligation method यह बताता है कि desired price, profit या loss की condition पाने के लिए items को किस ratio में मिलाना चाहिए।

यह concept business, trading और profit-based aptitude questions को जल्दी व सटीक तरीके से हल करने के लिए बहुत महत्वपूर्ण है।

Pattern: Profit-Based Mixture (Value Alligation)

Pattern

मुख्य विचार: Cost या selling prices को भी concentration की तरह treat करें - prices के अंतर से mixing का ratio मिलता है।

Steps:
1. Cheaper (C), Costlier (H) और Target/Mean (M) price पहचानें।
2. Differences निकालें: (H - M) और (M - C)।
3. Ratio of cheaper : costlier = (H - M) : (M - C)।
4. इस ratio का उपयोग desired total के लिए actual quantities निकालने में करें।

Step-by-Step Example

Question

एक व्यापारी ₹200/kg और ₹300/kg वाली दो तरह की चाय को मिलाता है। वह mixture को ₹260/kg पर बिना profit या loss बेचना चाहता है। उसे दोनों चाय किस ratio में मिलानी चाहिए?

Solution

  1. Step 1: Prices पहचानें

    Cheaper (C) = ₹200/kg, Costlier (H) = ₹300/kg, Mean price (M) = ₹260/kg.

  2. Step 2: Differences निकालें

    (H - M) = 300 - 260 = 40

    (M - C) = 260 - 200 = 60

  3. Step 3: Ratio बनाएँ

    Cheaper : Costlier = 40 : 60 = 2 : 3

  4. Step 4: Interpretation

    व्यापारी को cheaper और costlier tea को 2 : 3 के ratio में मिलाना चाहिए।

  5. Final Answer:

    Required ratio = 2 : 3 (cheaper : costlier)

  6. Quick Check:

    Weighted price = (2×200 + 3×300)/(2+3) = (400 + 900)/5 = ₹260 ✅

Quick Variations

1. अगर target price में profit % शामिल हो, तो पहले selling price निकालें फिर alligation लगाएँ।

2. Loss-based problems में mean price दोनों cost prices से कम होता है।

3. Discount दिए हों तो पहले effective selling price निकालें, फिर alligation apply करें।

Trick to Always Use

  • Step 1: Profit या loss percentages को पहले actual रुपए में बदलें।
  • Step 2: Quantity ratio पाने के लिए (Higher - Mean) : (Mean - Lower) का उपयोग करें।
  • Step 3: Weighted mean price निकालकर हमेशा verify करें।

Summary

Summary

Profit-Based Mixture (Value Alligation) pattern में:

  • Prices को concentration values की तरह treat करें।
  • Cheaper : Costlier ratio = (Higher - Mean) : (Mean - Lower).
  • Profit/loss percentages को पहले absolute price में convert करें।
  • Weighted average price निकालकर accuracy verify करें।

Practice

(1/5)
1. Tea costing ₹100/kg and ₹150/kg are mixed to produce a mixture worth ₹120/kg. Find the ratio of cheaper to costlier tea.
easy
A. 2 : 1
B. 1 : 2
C. 3 : 2
D. 2 : 3

Solution

  1. Step 1: Identify values

    Cheaper = ₹100, Costlier = ₹150, Mean = ₹120.

  2. Step 2: Apply value alligation

    (Higher - Mean) = 150 - 120 = 30; (Mean - Lower) = 120 - 100 = 20.

  3. Step 3: Form ratio (cheaper : costlier)

    Ratio = 30 : 20 = 3 : 2 → cheaper : costlier = 3 : 2.

  4. Final Answer:

    3 : 2 → Option C.

  5. Quick Check:

    Weighted mean = (3×100 + 2×150)/5 = (300 + 300)/5 = 600/5 = ₹120 ✅
Hint: Subtract mean from extremes and use those differences as ratio parts.
Common Mistakes: Reversing which difference corresponds to which component.
2. A shopkeeper mixes sugar costing ₹40/kg and ₹60/kg to get a mixture worth ₹50/kg. Find the ratio of the two types of sugar (cheaper : costlier).
easy
A. 1 : 1
B. 1 : 2
C. 1 : 3
D. 2 : 1

Solution

  1. Step 1: Identify costs

    Cheaper = ₹40, Costlier = ₹60, Mean = ₹50.

  2. Step 2: Apply alligation

    (H - M) = 60 - 50 = 10; (M - C) = 50 - 40 = 10.

  3. Step 3: Form ratio

    Cheaper : Costlier = 10 : 10 = 1 : 1.

  4. Final Answer:

    1 : 1 → Option A.

  5. Quick Check:

    (40 + 60)/2 = 50 ✅
Hint: Equal differences → equal parts (1 : 1).
Common Mistakes: Swapping which price corresponds to which difference.
3. Two varieties of pulses costing ₹80/kg and ₹100/kg are mixed to be sold at ₹96/kg at no profit, no loss. Find the ratio of cheaper to costlier variety.
easy
A. 1 : 3
B. 2 : 3
C. 3 : 2
D. 1 : 4

Solution

  1. Step 1: Identify data

    Cheaper = ₹80, Costlier = ₹100, Mean = ₹96.

  2. Step 2: Compute differences

    (H - M) = 100 - 96 = 4; (M - C) = 96 - 80 = 16.

  3. Step 3: Ratio (cheaper : costlier)

    4 : 16 = 1 : 4.

  4. Final Answer:

    1 : 4 → Option D.

  5. Quick Check:

    (80×1 + 100×4)/5 = (80 + 400)/5 = 480/5 = 96 ✅
Hint: Use (H - M) : (M - C) to get cheaper : costlier directly.
Common Mistakes: Forgetting to reverse differences when forming the ratio.
4. A trader mixes coffee costing ₹120/kg with another costing ₹180/kg and sells the mixture at ₹150/kg with 20% profit. Find the ratio of the two varieties (cheaper : costlier).
medium
A. 13 : 2
B. 11 : 1
C. 2 : 3
D. 5 : 6

Solution

  1. Step 1: Find mean cost (remove profit)

    Target selling price = ₹150 with 20% profit → mean cost M = 150 / 1.20 = ₹125.

  2. Step 2: Apply alligation

    (H - M) = 180 - 125 = 55; (M - C) = 125 - 120 = 5.

  3. Step 3: Form ratio (cheaper : costlier)

    55 : 5 = 11 : 1.

  4. Final Answer:

    11 : 1 → Option B.

  5. Quick Check:

    (11×120 + 1×180)/12 = (1320 + 180)/12 = 1500/12 = ₹125 (mean cost) → SP with 20% profit = 125×1.2 = ₹150 ✅
Hint: Convert SP and profit% to effective cost (SP/(1+profit%)) before alligation.
Common Mistakes: Using selling price directly as mean without removing profit.
5. A milkman mixes milk costing ₹30/litre with water costing ₹0/litre to get a mixture worth ₹24/litre. Find the ratio of milk to water.
medium
A. 4 : 1
B. 3 : 2
C. 2 : 1
D. 5 : 1

Solution

  1. Step 1: Identify data

    Milk = ₹30/litre, Water = ₹0/litre, Mean = ₹24/litre.

  2. Step 2: Apply alligation

    (H - M) = 30 - 24 = 6; (M - C) = 24 - 0 = 24.

  3. Step 3: Form ratio (water : milk)

    6 : 24 = 1 : 4, so milk : water = 4 : 1.

  4. Final Answer:

    4 : 1 → Option A.

  5. Quick Check:

    (4×30 + 1×0)/5 = 120/5 = ₹24 ✅
Hint: Water has zero cost - the difference gives the parts of milk directly.
Common Mistakes: Confusing order milk : water vs water : milk when interpreting ratio.

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