Mixtures, Alligation and Replacement Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Mixtures, Alligation and Replacement notes for CAT: 70 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Mixtures, Alligation and Replacement

CAT Quant • Arithmetic

Mixtures, Alligation and Replacement

A chapter about average value, concentration changes, and transfer logic.

chapter PYQs

⚖️

t1 — Alligation, Blending and Weighted Cost

Master weighted average, two-price mixing, ingredient percentage, and profit-linked blends.

Why first? This is the foundation of the whole chapter.

7 PYQs

High value

🔁

t2 — Successive Replacement and Dilution

Handle remove-and-refill cases using repeated fraction left.

8 PYQs

Very likely

🧪

t3 — Two-Vessel Transfer Problems

Track what moves between two containers without losing total quantity logic.

3 PYQs

Selective

End goal: When CAT gives prices, percentages, ratios, or ingredients, you should immediately ask: what is the weighted average?

Topic Hero: Alligation, Blending and Weighted Cost

Selected Topic • t1

Alligation, Blending and Weighted Cost

The art of mixing values without doing long equations every time.

⚖️

Core idea

Final value is a weighted average.

₹

CAT use

Cost price, selling price, profit, and marked price blends.

Ingredient use

Coffee, cocoa, sugar, silver, copper, milk, syrup.

One-line hook

If the mixture average is closer to one item, that item must be used more.

The Real Meaning of a Mixture

A mixture is controlled by quantity

When two items are mixed, the final value depends on both:

Value

price / percentage / strength

Weight

quantity used

Core formula:

Average value = \frac{total value}{total quantity}

Example: If ₹300/kg tea and ₹800/kg tea are mixed, the final price must be between ₹300 and ₹800. It cannot be below ₹300 or above ₹800 unless extra profit/loss is involved.

Weighted Average Formula

The master formula

Weighted average = \frac{q _{1} v _{1} + q _{2} v _{2} + \dots + q _{n} v _{n}}{q _{1} + q _{2} + \dots + q _{n}}

Symbol	Meaning
$q_{i}$	quantity of the item
$v_{i}$	value per unit: price, percentage, concentration, strength

Exam memory: First multiply, then add, then divide. Never average the rates directly unless quantities are equal.

Mixtures, Alligation and Replacement: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Ability NAT

A trader buys $100$ kg of sugar at Rs $40$ /kg and $60$ kg of jaggery at Rs $60$ /kg. He sells $40$ kg of the sugar separately at a $10%$ profit. He then mixes the remaining sugar with all the jaggery. To achieve an overall profit of $25%$ on the entire original purchase, at what price per kg must he sell the mixture?

Correct Answer:

64.5

Step-by-Step Solution

Key idea: This is a partial sale + mixture profit problem. The trap is averaging selling prices. We must track Total Cost Price (CP) and Required Total Selling Price (SP).

Step 1: Calculate Total CP.

Sugar CP = $100 \times 40 = 4000$ .

Jaggery CP = $60 \times 60 = 3600$ .

Total CP = $4000 + 3600 = 7600$ .

Step 2: Calculate Required Total SP.

Overall Profit = $25%$ .

Required Total SP = $7600 \times 1.25 = 9500$ .

Step 3: Account for Revenue Already Earned.

Sold Sugar = $40$ kg. SP/kg = $40 \times 1.10 = 44$ .

Revenue from Sugar = $40 \times 44 = 1760$ .

Step 4: Determine Required Revenue from Mixture.

Remaining Revenue Needed = $9500 - 1760 = 7740$ .

Step 5: Find Mixture Selling Price.

Mixture Composition:

Remaining Sugar = $100 - 40 = 60$ kg.

Jaggery = $60$ kg.

Total Mixture Weight = $60 + 60 = 120$ kg.

Required SP/kg = $\frac{7740}{120} = 64.5$ .

Answer: 64.5

Question 2 · Quantitative Ability NAT

A trader buys $80$ kg of dry fruit A and $120$ kg of dry fruit B.

Per kg, A costs $25%$ more than B.

Before selling, $10%$ of A and $20%$ of B are found damaged and cannot be sold.

The remaining undamaged stock is mixed and sold at Rs $132$ per kg.

If the trader makes an overall profit of $20%$ on the entire purchase, what is the cost price of B, in rupees per kg?

Correct Answer:

Step-by-Step Solution

Key idea: this is an overall-profit mixture question with spoilage, recognisable because the selling revenue comes only from undamaged stock, but the profit percentage is calculated on the whole purchase cost.

Step 1: Let the cost price of B be $b$ rupees per kg.

Since A costs $25%$ more than B,

$cost of A = 1.25 b .$

Step 2: Compute the total cost price of the entire purchase.

Cost of A:

$80 \times 1.25 b = 100 b .$

Cost of B:

$120 \times b = 120 b .$

Total cost:

$100 b + 120 b = 220 b .$

Step 3: Find the undamaged quantities.

Damaged A:

$10% of 80 = 8 kg .$

Undamaged A:

$80 - 8 = 72 kg .$

Damaged B:

$20% of 120 = 24 kg .$

Undamaged B:

$120 - 24 = 96 kg .$

Total sellable mixture:

$72 + 96 = 168 kg .$

Step 4: Compute total selling revenue.

The mixture is sold at Rs $132$ per kg:

$S P = 168 \times 132.$

Calculate:

$168 \times 132 = 168 (100 + 30 + 2) = 16800 + 5040 + 336 = 22176.$

Step 5: Use the overall profit condition.

Overall profit is $20%$ on the entire purchase, so

$S P = 1.2 \times C P .$

Therefore

$22176 = 1.2 \times 220 b .$

Compute:

$1.2 \times 220 b = 264 b .$

$264 b = 22176.$

Step 6: Solve for $b$ .

$b = \frac{22176}{264} = 84.$

Answer: 84.

More notes in this unit

chapter

Mixtures, Alligation and Replacement Notes for CAT: Concepts, Formulas, Worked Examples & Practice

Mixtures, Alligation and Replacement notes for CAT: 70 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

A question from this chapter

Question 1

A trader buys 100 kg of sugar at Rs 40 /kg and 60 kg of jaggery at Rs 60 /kg. He sells 40 kg of the sugar separately at a 10% profit. He then mixes the remaining sugar with all the jaggery. To achieve an overall profit of 25% on the entire original purchase, at what price per kg must he sell the mixture?

Question 2

A trader buys 80 kg of dry fruit A and 120 kg of dry fruit B. Per kg, A costs 25% more than B. Before selling, 10% of A and 20% of B are found damaged and cannot be sold. The remaining undamaged stock is mixed and sold at Rs 132 per kg. If the trader makes an overall profit of 20% on the entire purchase, what is the cost price of B, in rupees per kg?

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