Mixture and Alligation Made Practical | Banking Quant Mastery - Study Chapter | QuizMaker
Use weighted averages and alligation to solve value-mixing questions quickly.
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Course
Banking Quant Mastery: Arithmetic to Data Sufficiency
Topic
Module 2: Commercial Arithmetic and Value Judgement
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
| Concept | Formula / Rule |
|---|---|
| Mean value | mean=total valuetotal quantity\text{mean}=\frac{\text{total value}}{\text{total quantity}}mean=total quantitytotal value |
| Alligation ratio | cheaper:dearer=(dearer−mean):(mean−cheaper)\text{cheaper} : \text{dearer}=(\text{dearer}-\text{mean}):(\text{mean}-\text{cheaper})cheaper:dearer=(dearer−mean):(mean−cheaper) |
| Replacement after n operations | pure liquid left=x(1−yx)n\text{pure liquid left}=x\left(1-\frac{y}{x}\right)^npure liquid left=x(1−xy)n |
How To Approach Questions
- Mark the cheaper value, dearer value, and mean value.
- Use cross-differences for the required ratio.
- For replacement, multiply by the remaining fraction after each operation.
- If the ratio changes after adding water, first convert the old ratio into actual litres.
Worked Examples
Example 1
Prompt: Rice worth Rs 40\text{Rs }40Rs 40 per kg is mixed with rice worth Rs 60\text{Rs }60Rs 60 per kg to get a mean price of Rs 48\text{Rs }48Rs 48. Find the ratio.
Approach: Alligation gives (60−48):(48−40)=12:8=3:2(60-48):(48-40)=12:8=3:2(60−48):(48−40)=12:8=3:2. So the ratio of cheaper to dearer rice is 3:23:23:2.
Example 2
Prompt: A vessel contains 404040 litres of milk. If 101010 litres are removed and replaced with water, what fraction of milk remains?
Approach: Remaining milk fraction =1−1040=34=1-\frac{10}{40}=\frac34=1−4010=43. So milk left =40×34=30=40\times\frac34=30=40×43=30 litres.
Example 3
Prompt: A vessel contains 100100100 litres of spirit. If 101010 litres are removed and replaced with water three times, how much spirit remains?
Approach: Use the replacement formula: spirit left =100(1−10100)3=100(910)3=72.9=100\left(1-\frac{10}{100}\right)^3=100\left(\frac9{10}\right)^3=72.9=100(1−10010)3=100(109)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.
Tags
- banking quant
- mixture
- alligation
- weighted average