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Profit-Based Mixture (Value Alligation)

Introduction

Many mixture problems involve items bought at different costs or sold at different prices and then mixed or resold. The Value Alligation method helps to find in what ratio the items should be mixed to achieve a desired price, profit, or loss condition.

This concept is essential for solving business, trading, and profit-related aptitude problems quickly and accurately.

Pattern: Profit-Based Mixture (Value Alligation)

Pattern

The key idea: Treat cost or selling prices like concentrations - the differences between prices give the ratio of quantities to mix.

Steps to follow:
1. Identify Cheaper (C), Costlier (H), and Target/Mean (M) price.
2. Find the differences: (H - M) and (M - C).
3. The ratio of cheaper : costlier = (H - M) : (M - C).
4. Use this ratio to find actual quantities for any desired total.

Step-by-Step Example

Question

A merchant mixes tea costing ₹200/kg and ₹300/kg. He wants to sell the mixture at ₹260/kg without profit or loss. In what ratio should he mix the two types of tea?

Solution

  1. Step 1: Identify prices

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

  2. Step 2: Compute differences

    (H - M) = 300 - 260 = 40; (M - C) = 260 - 200 = 60.

  3. Step 3: Form ratio

    Cheaper : Costlier = (H - M) : (M - C) = 40 : 60 = 2 : 3.

  4. Step 4: Interpretation

    The merchant should mix the cheaper and costlier teas in the ratio 2 : 3.

  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. When the target price includes desired profit %, compute the selling price first, then apply alligation.

2. For loss-based problems, the mean price will be lower than both cost prices.

3. If discounts are given, adjust selling prices to effective selling prices before applying alligation.

Trick to Always Use

  • Step 1: Convert all profit or loss percentages into actual rupee values first.
  • Step 2: Use (Higher - Mean) : (Mean - Lower) to get the quantity ratio.
  • Step 3: Always verify by calculating the weighted mean price.

Summary

Summary

In the Profit-Based Mixture (Value Alligation) pattern:

  • Treat prices as values for alligation just like concentrations.
  • The ratio of cheaper : costlier = (Higher - Mean) : (Mean - Lower).
  • Convert profit/loss percentages to absolute prices before applying the formula.
  • Always cross-verify by recalculating the weighted average price for accuracy.

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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