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Mixture/Alligation in % Form

Introduction

Some aptitude questions involve mixing two or more items (like milk and water, alcohol and water, rice varieties, etc.) with different percentages.

These are called mixture or alligation problems. They test your ability to combine percentages and find the final concentration, or the ratio of mixing.

Pattern: Mixture/Alligation in % Form

Pattern

Average % = (Quantity₁ × %₁ + Quantity₂ × %₂) ÷ (Quantity₁ + Quantity₂)

Alligation rule: (High% - Mean%) : (Mean% - Low%) = Ratio of mixing

Step-by-Step Example

Question

A mixture contains 30% milk and the rest water. Another mixture contains 70% milk. In what ratio should the two mixtures be mixed so that the resulting mixture has 50% milk?

Solution

  1. Step 1: Identify the percentages.

    First mixture = 30% milk (low %)
    Second mixture = 70% milk (high %)
    Final mixture = 50% milk (mean %)

  2. Step 2: Apply alligation rule.

    Sentence: Use (High - Mean) : (Mean - Low)
    Math: (70 - 50) : (50 - 30) = 20 : 20

  3. Step 3: Simplify the ratio.

    Ratio = 1 : 1

  4. Step 4: Final Answer.

    The two mixtures should be mixed in the ratio 1 : 1.

  5. Step 5: Quick Check (average method).

    (30 + 70) ÷ 2 = 50% ✅

Quick Variations

If quantities are given: Use weighted average formula. Eg: 2 liters of 40% sugar solution mixed with 3 liters of 60% → (2×40 + 3×60) ÷ (2+3) = 52%.

If one solution is pure: Treat pure milk as 100% and pure water as 0% in alligation.

Trick to Always Use

  • Step 1: Identify low %, high %, and mean %.
  • Step 2: Apply (High - Mean) : (Mean - Low).
  • Step 3: This gives the mixing ratio.
  • Step 4: If quantities are given, use weighted average formula.

Summary

Summary

The Mixture/Alligation pattern is solved using either the weighted average formula or the alligation rule.

Alligation formula: (High - Mean) : (Mean - Low)

  • High % = stronger concentration.
  • Low % = weaker concentration.
  • Mean % = desired concentration.

With practice, you’ll quickly solve mixture questions using the shortcut ratio method.

Practice

(1/5)
1. A solution contains 20% sugar. If you take 1 liter of this solution, how much sugar is in it?
easy
A. 0.2 liters
B. 0.25 liters
C. 0.15 liters
D. 0.3 liters

Solution

  1. Step 1: Identify total and percentage

    Total solution = 1 liter, Sugar% = 20%.

  2. Step 2: Convert percent to fraction and multiply

    Sugar = (20 ÷ 100) × 1 = 0.2 liters.

  3. Final Answer:

    0.2 liters → Option A

  4. Quick Check:

    10% of 1 liter = 0.1 → 20% = 0.2 ✅
Hint: Sugar = (percentage ÷ 100) × total quantity.
Common Mistakes: Using 20 directly instead of converting to fraction of total.
2. A solution of 40% alcohol is mixed with 60% alcohol solution in equal quantities. What is the percentage of alcohol in the mixture?
easy
A. 48%
B. 50%
C. 52%
D. 54%

Solution

  1. Step 1: Note concentrations and ratio

    Two solutions: 40% and 60%, mixed in 1:1 ratio.

  2. Step 2: Take the simple average for equal quantities

    Average % = (40 + 60) ÷ 2 = 50%.

  3. Final Answer:

    50% → Option B

  4. Quick Check:

    Middle of 40% and 60% = 50% ✅
Hint: Equal quantities → just take simple average.
Common Mistakes: Forgetting that equal quantities means direct average.
3. In what ratio should a 20% acid solution and a 60% acid solution be mixed to get a 40% acid solution?
medium
A. 1:1
B. 2:1
C. 3:1
D. 1:2

Solution

  1. Step 1: Record low, high and mean percentages

    Low% = 20, High% = 60, Mean% = 40.

  2. Step 2: Apply alligation

    (High - Mean) : (Mean - Low) = (60 - 40) : (40 - 20) = 20 : 20.

  3. Final Answer:

    1:1 → Option A

  4. Quick Check:

    Equal mix of 20% and 60% → average = 40% ✅
Hint: Use alligation: (High-Mean):(Mean-Low).
Common Mistakes: Subtracting in wrong order or mixing up which side is high/low.
4. Two sugar solutions of 30% and 50% are mixed in the ratio 3:2. What is the percentage of sugar in the final mixture?
medium
A. 36%
B. 38%
C. 40%
D. 42%

Solution

  1. Step 1: Note quantities ratio and concentrations

    Quantities ratio = 3:2 → Total = 5 parts.

  2. Step 2: Compute weighted average

    Weighted average = (3×30 + 2×50) ÷ 5 = (90 + 100) ÷ 5 = 190 ÷ 5 = 38%.

  3. Final Answer:

    38% → Option B

  4. Quick Check:

    Value lies between 30% and 50% closer to 30% (since more of 30%) ✅
Hint: Weighted average = (Σ quantity × %) ÷ total quantity.
Common Mistakes: Taking simple average instead of weighted average.
5. A container has 25% alcohol solution. Another container has 45% alcohol solution. In what ratio should they be mixed to obtain 30% solution?
medium
A. 2:3
B. 3:2
C. 3:1
D. 4:1

Solution

  1. Step 1: Note low, high and desired percentages

    Low% = 25, High% = 45, Mean% = 30.

  2. Step 2: Apply alligation

    Alligation ratio = (45 - 30):(30 - 25) = 15:5 = 3:1.

  3. Final Answer:

    3:1 → Option C

  4. Quick Check:

    Mix 3 parts of 25% with 1 part of 45% → weighted average = (3×25 + 1×45)/4 = (75 + 45)/4 = 120/4 = 30% ✅
Hint: Alligation rule → (High-Mean):(Mean-Low).
Common Mistakes: Confusing ratio order (low vs high concentration).

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