How to Handle Mixtures and Alligations Questions in CAT Quant 2025

Mixture vs Alligation

Mixture is when two or more elements / compounds are mixed together in certain quantities / ratio.

Alligation is a rule which will help to find the ration in which two or more elements / compounds are mixed together.

Type of Question

Explanation

Questions

Simple Mixture Problems

• Mixing two or more types of items (milk and water, sugar solutions, rice of different prices, etc.).
• Direct application of Alligation rule to find the ratio in which they should be mixed.

In what ratio should tea worth Rs 60/kg be mixed with tea worth Rs 90/kg so that the mixture costs Rs 75/kg?

Replacement (Repeated Dilution) Problems

A container has a mixture of two liquids. A part is removed and replaced with another liquid, and this process is repeated multiple times.

A vessel contains 60 L of milk. 6 L is replaced with water and this is repeated 3 times. Find the final quantity of milk.

Value Mixture Problems (Cost/Concentration)

• Mixing items with different prices or solutions with different concentrations.
• Used to find mean price / average concentration.

In what ratio should a 20% alcohol solution be mixed with a 50% solution to get a 30% solution?

Successive Mixtures

• When mixtures are formed step by step (first two are mixed, then another is added).
• Often solved by applying weighted average repeatedly or directly by alligation.

A solution of milk and water in 3:2 is mixed with another solution of 5:3 in the ratio 4:1. Find the final ratio.

Profit and Loss Based Mixture Problems

• Shopkeeper mixes two varieties of goods and sells at a price, earning a profit or loss.
• Often requires combining Mixtures & Alligation with Profit and Loss concepts.

A dishonest milkman mixes water with milk and sells at cost price, making 25% profit. What is the ratio of milk to water?

Replacement with Different Substance (Not Just Water)

• Instead of water, another mixture or liquid is added.
• Slightly more complex than simple replacement.

A container has alcohol and water in 7:5. 10 L is removed and replaced with pure alcohol. Find the new ratio.

Advanced Puzzle-based Mixture Problems (CAT/DILR flavored)

• Sometimes appear in Quant-DILR hybrid sets, where multiple conditions are given on mixing items/solutions.
• Require step-by-step logical deduction rather than direct formula.

Flowchart / Diagram

Purpose

Tip

Basic Alligation Diagram (Rule of Mixture)

To find the ratio in which two or more ingredients at different prices must be mixed to get a desired price.

• Always subtract "Cheaper from Mean" and "Mean from Dearer" to avoid confusion.
• Use cross-subtraction: D – M : M – C

Replacement/Repeated Alligation Flowchart

Used when a part of a mixture is replaced by another ingredient multiple times.

• For successive replacements, multiply the remaining fraction each time.
• If 20% is replaced, then 80% remains.

Table for Weighted Average (Mixture of More than Two Elements)

When mixing more than two ingredients, use a weighted average table

If quantities are not given, assume variables or use ratios.

Profit/Loss in Mixture Problems (Flowchart)

To find profit or loss when selling a mixture at a different rate.

If a shopkeeper mixes two varieties, first find the mean cost price before calculating profit.

Alligation for Speed, Time Work Problems

Used in problems like average speed or work efficiency.

Works similarly for work efficiency (e.g., two workers with different efficiencies).

Common Mistakes

Detailed Errors

Solution

Misunderstanding the Basics of Alligation

Students confuse alligation rule with simple averaging or fail to identify the correct components (cheaper, dearer, or resultant mixture).

• Remember: Alligation is used to find the ratio in which two or more ingredients are mixed to produce a mixture at a given price or concentration.
• Clearly identify:
• Cheaper quantity (C) – Lower value (price/percentage).
• Dearer quantity (D) – Higher value (price/percentage).
• Mean price (M) – Desired mixture value.

Incorrectly Applying the Alligation Formula

Students misplace values in the formula

• Always ensure:
• (D – M) is the difference between dearer and mean.
• (M – C) is the difference between mean and cheaper.

Ignoring Units or Concentration Consistency

Mixing different units (e.g., % and absolute values) without conversion.

• Ensure all quantities are in the same unit (e.g., convert % to actual values if needed).
• Example: If a problem involves 20% alcohol and 30% alcohol, express them as 0.2 and 0.3 when working with fractions.

Misinterpreting "Water Added" Problems

Treating water as a 0% component incorrectly.

Water has 0% concentration of the solute (e.g., alcohol, milk)

Overlooking Profit / Loss in Mixtures

Forgetting to account for profit/loss when calculating selling price.

If a mixture is sold at a profit/loss, adjust the mean price accordingly.

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