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Basic Mixture Concept

Introduction

Mixture problems involve combining two or more components with different concentrations, costs, or qualities. This pattern is essential because it builds the foundation for all other mixture problems, including alligation and replacement cases.

Understanding how to find the final mixture concentration using weighted average helps in both mathematical and real-world applications.

Pattern: Basic Mixture Concept

Pattern

The key idea: The final concentration is the weighted average of individual concentrations based on their quantities.

Formula used:
Final Concentration = (Sum of all pure component amounts) ÷ (Total quantity)

Steps:
1. Convert % to decimal (if required)
2. Multiply each quantity by its concentration
3. Add contributions and divide by total volume

Step-by-Step Example

Question

You have 30 litres of a 20% acid solution and 10 litres of a 50% acid solution. Find the final concentration of acid in the mixture.

Solution

  1. Step 1: Identify given data

    30 L @ 20% and 10 L @ 50%.

  2. Step 2: Convert percentages to decimals

    20% = 0.20, 50% = 0.50.

  3. Step 3: Compute pure acid content

    Find the amount of pure acid in each:

    • From first: 30 × 0.20 = 6 L
    • From second: 10 × 0.50 = 5 L

  4. Step 4: Combine acid and total mixture volume

    Total acid = 6 + 5 = 11 L; total volume = 30 + 10 = 40 L.

  5. Step 5: Calculate final concentration

    Final concentration = (11 ÷ 40) × 100 = 27.5%.

  6. Final Answer:

    27.5%

  7. Quick Check:

    The final % (27.5%) lies between 20% and 50% and is closer to 20% (since that quantity was larger). ✅

Quick Variations

1. If one liquid is pure (100%), use the same formula - its contribution equals its full volume.

2. Works the same for solids (e.g., salt + sand mixtures).

3. When one quantity is water (0%), it simply dilutes the concentration.

Trick to Always Use

  • Step 1 → Convert percentages to decimals (or use % consistently).
  • Step 2 → Multiply quantity × concentration for each component.
  • Step 3 → Add results and divide by total quantity.
  • Step 4 → Multiply by 100 to express as percentage.

Summary

Summary

In the Basic Mixture Concept pattern:

  • Final concentration is a weighted average of the component concentrations.
  • Always multiply each part’s quantity by its concentration to find pure content.
  • Divide the total pure content by total quantity to get final strength.
  • Quick check: The final concentration must lie between the two given concentrations.

Practice

(1/5)
1. A solution of 20 litres contains 25% sugar. How much pure sugar is in the solution?
easy
A. 5 litres
B. 4 litres
C. 6 litres
D. 3 litres

Solution

  1. Step 1: Identify Data

    Total solution = 20 L; concentration = 25%.

  2. Step 2: Convert to Decimal

    25% = 0.25.

  3. Step 3: Calculate Pure Sugar

    Pure sugar = 20 × 0.25 = 5 litres.

  4. Final Answer:

    Pure sugar = 5 litres → Option A.

  5. Quick Check:

    5 ÷ 20 = 0.25 → 25% ✅
Hint: Multiply total quantity by concentration fraction.
Common Mistakes: Dividing instead of multiplying or skipping % to decimal conversion.
2. A 40-litre mixture of milk and water contains 30% milk. If 10 litres of milk is added, find the new percentage of milk.
easy
A. 44%
B. 45%
C. 47.5%
D. 50%

Solution

  1. Step 1: Find Original Milk Quantity

    Milk = 40 × 0.30 = 12 L.

  2. Step 2: Add Extra Milk

    Added milk = 10 L → Total milk = 12 + 10 = 22 L.

  3. Step 3: Find New Total Volume

    Total mixture = 40 + 10 = 50 L.

  4. Step 4: Calculate New Concentration

    (22 ÷ 50) × 100 = 44%.

  5. Final Answer:

    New concentration = 44% → Option A.

  6. Quick Check:

    22 ÷ 50 = 0.44 → 44% ✅
Hint: Add pure quantity directly, then divide by new total.
Common Mistakes: Adding 10% instead of 10 litres of milk.
3. Two salt solutions of 10 litres each have concentrations 20% and 40%. What is the concentration after mixing them?
easy
A. 25%
B. 28%
C. 30%
D. 35%

Solution

  1. Step 1: Find Pure Salt in Each Solution

    First: 10 × 0.20 = 2 L; Second: 10 × 0.40 = 4 L.

  2. Step 2: Total Pure Salt

    2 + 4 = 6 L.

  3. Step 3: Total Volume

    10 + 10 = 20 L.

  4. Step 4: Find Final Concentration

    (6 ÷ 20) × 100 = 30%.

  5. Final Answer:

    Final concentration = 30% → Option C.

  6. Quick Check:

    Equal volumes → average of 20% and 40% = 30% ✅
Hint: For equal quantities, the final % is the average of both concentrations.
Common Mistakes: Using average formula when volumes differ.
4. A 15-litre solution has 6 litres of alcohol. What percentage of alcohol is in the solution?
medium
A. 30%
B. 35%
C. 45%
D. 40%

Solution

  1. Step 1: Identify Known Values

    Pure alcohol = 6 L; total solution = 15 L.

  2. Step 2: Apply Percentage Formula

    (6 ÷ 15) × 100 = 40%.

  3. Step 3: Interpret Result

    40% means 40 parts of every 100 are alcohol.

  4. Final Answer:

    Alcohol percentage = 40% → Option D.

  5. Quick Check:

    15 × 0.40 = 6 L → consistent ✅
Hint: Use (part ÷ total) × 100 to get percentage.
Common Mistakes: Dividing total by part instead of part by total.
5. A container has 25 litres of a 60% sugar solution. How much water must be added to make it a 40% solution?
medium
A. 10 litres
B. 12.5 litres
C. 15 litres
D. 20 litres

Solution

  1. Step 1: Find Initial Pure Sugar

    25 × 0.60 = 15 L of sugar.

  2. Step 2: Assume x Litres of Water Added

    New total volume = 25 + x.

  3. Step 3: Set Up Equation for New Concentration

    (15 ÷ (25 + x)) × 100 = 40.

  4. Step 4: Solve for x

    15 = 0.40 × (25 + x) → 15 = 10 + 0.4x → x = 12.5 L.

  5. Final Answer:

    Water to be added = 12.5 litres → Option B.

  6. Quick Check:

    Total = 37.5 L; 15 ÷ 37.5 = 0.4 → 40% ✅
Hint: Pure content stays constant; solve using (pure ÷ total) = target fraction.
Common Mistakes: Assuming total amount remains unchanged after adding water.

Mock Test

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