SSC CGL Preparation – Day 13

Quantitative Aptitude: Mixtures and Alligations

---

🔍 What is a Mixture?

A mixture is formed by mixing two or more ingredients/substances. It could be two types of liquids, or solids and liquids, like milk & water, acid & water, or two varieties of rice/sugar.

---

🧠 Key Concepts

1. Mean Price

When two or more components are mixed, the mean price is the average cost price of the mixture.

Formula: $$\text{Mean Price} = \frac{\text{Total Cost}}{\text{Total Quantity}}$$

---

2. Alligation Rule (Cross Method)

The Alligation rule is a quick method to determine the ratio in which two items at different prices must be mixed to obtain a mixture at a given price.

📌 Alligation Formula:

If two items at prices P₁ and P₂ are mixed to get a mixture at price M: $$\text{Ratio of quantities} = \frac{M – P_2}{P_1 – M}$$

Where:

• P₁ = cost price of dearer (more expensive) ingredient
• P₂ = cost price of cheaper ingredient
• M = mean price (mixture price)

---

📚 Types of Mixture Problems

---

🔹 Type 1: Mixing Two Quantities of Different Costs

Example 1:
Tea costing ₹60/kg and ₹90/kg are mixed to get tea worth ₹80/kg. What is the ratio?

Solution: $$\frac{90 – 80}{80 – 60} = \frac{10}{20}$$

✅ Mix cheaper (₹60) and costlier (₹90) in ratio 1:2

---

🔹 Type 2: Replacing Part of Mixture

When part of the mixture is removed and replaced with another substance (like water), we use the formula: $$\text{Final Quantity of Original Substance} = Q \left(1 – \frac{R}{Q} \right)^n$$

Where:

• Q = total quantity
• R = replaced quantity each time
• n = number of operations

Example 2:
A container has 40L milk. 8L is removed and replaced with water. This process is repeated 2 more times. Find the final amount of milk. $$= 40 \left(1 – \frac{8}{40} \right)^3 = 40 \times \left(\frac{4}{5}\right)^3 = 40 \times \frac{64}{125} = 20.48L$$

✅ Milk left = 20.48L

---

🔹 Type 3: Profit-based Mixture Problems

Used when one component is water (free) and the seller wants to earn a profit.

Formula: $$\text{Cheater’s Ratio (Water:Milk)} = \frac{\text{Profit %}}{100 + \text{Profit %}}$$

Example 3:
A dishonest milkman adds water to milk and sells the mixture at CP of milk. If he wants a 20% profit, how much water should be added to 1L of milk? $$\text{Water:Milk} = \frac{20}{120} = \frac{1}{6}$$

✅ Add 1/6 L = 166.67 ml water

---

✍️ Important Shortcuts & Tricks

Type	Trick
Alligation	Cross multiplication gives direct ratio
Replacement	Use exponential formula
Profit mix	Use Profit / (100 + Profit) for water-to-milk ratio
Ratio finding	Use alligation formula when mixture price is known

---

🧪 Practice Examples

---

Q1. In what ratio should sugar worth ₹36/kg be mixed with sugar worth ₹42/kg so that the mixture is worth ₹40/kg?

✅ Solution: $$\frac{42 – 40}{40 – 36} = \frac{2}{4} = 1:2$$

---

Q2. A 60L mixture contains milk and water in 2:1 ratio. 15L of this is removed and replaced with water. What is the new ratio?

Step 1: Initial milk = (2/3) × 60 = 40L, water = 20L
Step 2: In 15L removed, milk = 10L, water = 5L
Step 3: After replacement:

• Milk = 40 − 10 = 30L
• Water = 20 − 5 + 15 = 30L
  ✅ New Ratio = 1:1

---

Q3. 2 types of rice costing ₹30/kg and ₹40/kg are mixed in 3:2 ratio. Find the price per kg of the mixture. $$= \frac{(3 × 30) + (2 × 40)}{3 + 2} = \frac{90 + 80}{5} = ₹34/kg$$

---

📌 Common SSC CGL Questions

• Use of alligation to find ratio
• Repeated replacement formula
• Water-milk profit cheat formula
• Marked price vs cost price in mixture questions

---

🏁 Summary

Concept	Formula
Alligation Ratio	$$(C₁ – Mean):(Mean – C₂)$$
Repeated Replacement	$$Q \times (1 – R/Q)^n$$
Water-Milk Profit Ratio	$$\text{Profit}/(100 + \text{Profit})$$

---

📝 Assignment for Practice

1. A merchant mixes 2 varieties of tea worth ₹50 and ₹80 per kg in a ratio to make a mixture worth ₹70/kg. Find the ratio.
2. 30L of a mixture has milk and water in 7:3 ratio. 6L is removed and replaced with water. What is the new ratio?
3. A dishonest milkman adds water to earn 25% profit. Find the ratio of water to milk.