10. Mixture and Alligation Made Practical

Why This Chapter Matters

Mixture and alligation questions appear simple but hide many formats: mean price, dilution, replacement, adulteration, and ratio shifts after adding water or another liquid. Once the mean and remaining-fraction logic is clear, the whole chapter speeds up.

Core Ideas

• When two items with different prices or concentrations are mixed, the result lies between the two source values.
• Alligation gives the quantity ratio by cross-differences around the mean.
• Replacement questions change the composition repeatedly, so track the remaining fraction each time.
• In adulteration questions, water has zero cost, so alligation becomes a quick way to model profit-at-cost-price tricks.

High-Value Formulas

How To Approach Questions

1. Mark the cheaper value, dearer value, and mean value.
2. Use cross-differences for the required ratio.
3. For replacement, multiply by the remaining fraction after each operation.
4. If the ratio changes after adding water, first convert the old ratio into actual litres.

Worked Examples

Example 1

Prompt: Rice worth $\text{Rs }40$ per kg is mixed with rice worth $\text{Rs }60$ per kg to get a mean price of $\text{Rs }48$. Find the ratio.

Approach: Alligation gives $(60-48):(48-40)=12:8=3:2$. So the ratio of cheaper to dearer rice is $3:2$.

Example 2

Prompt: A vessel contains $40$ litres of milk. If $10$ litres are removed and replaced with water, what fraction of milk remains?

Approach: Remaining milk fraction $=1-\frac{10}{40}=\frac{3}{4}$. So milk left $=40\times \frac{3}{4}=30$ litres.

Example 3

Prompt: A vessel contains $100$ litres of spirit. If $10$ litres are removed and replaced with water three times, how much spirit remains?

Approach: Use the replacement formula: spirit left $=100{\left(1-\frac{10}{100}\right)}^3=100{\left(\frac{9}{10}\right)}^3=72.9$ litres.

Common Mistakes

• Reversing the ratio in alligation.
• Forgetting that the result must lie between the two source values.
• Using replacement logic without updating the remaining fraction.
• Treating a water-addition question like a simple subtraction problem when the total mixture volume also changes.

Quick Revision

Think weighted average first, then ask whether the problem is about pricing, dilution, or repeated replacement.