मिश्रण और अलिगेशन
Mixtures and Alligation
Alligation is a shortcut that turns weighted-average and mixture questions into a fast cross calculation, and it is one of the highest-return tricks in the whole arithmetic block. SBI Clerk uses it for mixing two prices, two concentrations, or two speeds, and for the classic 'replace part of a mixture' question. The rule finds the ratio in which two ingredients at different values are mixed to hit a target average, and once the cross method is automatic these questions fall in twenty seconds.
The Alligation Rule
Place the two given values at the top corners and the mean value in the middle, then take the positive difference along each diagonal. The ratio of cheaper to dearer quantity equals (dearer − mean) to (mean − cheaper). In words, the quantity of each ingredient is proportional to the distance of the other ingredient's value from the mean. This cross subtraction replaces setting up and solving a weighted-average equation, and it works for price, purity, concentration or any quantity that averages linearly.
Cheaper : Dearer = (Dearer value − Mean) : (Mean − Cheaper value)
Reading the Cross Method Right
The answer of alligation is always a ratio of quantities, not the quantities themselves, so a follow-up step is usually needed to reach actual amounts. Keep the labels straight: the difference nearer the cheaper value belongs to the dearer quantity, and vice versa, because each quantity is pulled toward the mean by the other. A quick sense check is that the ingredient whose value is closer to the mean must be present in the larger quantity, which catches a flipped ratio before it costs you the mark.
| Use case | The two values |
|---|---|
| Mixing two priced items | Cost per unit of each |
| Diluting or strengthening | Concentration of each |
| Average speed of a journey | The two speeds |
| Profit on a blend | Cost or selling price of each |
The Replacement Formula
A staple question removes part of a solution and replaces it with water, repeatedly, and asks the final quantity of the original liquid. After n replacements, the fraction remaining is (1 − replaced/total) raised to the power n, times the original amount. If 10 litres of the 100-litre content is drawn and replaced with water twice, the pure liquid left is 100 × (9/10) × (9/10) = 81 litres. This one formula handles every 'repeated replacement' question without tracking each step by hand.
Final pure quantity = Initial × (1 − replaced/total)^n for n replacements
Solved Examples
Example 1: In what ratio mix rice at 30 and 45 per kg to sell a blend at 35? By alligation, (45 − 35) : (35 − 30) = 10 : 5 = 2 : 1, cheaper to dearer.
Example 2: From 80 litres of milk, 8 litres are replaced by water twice. Milk left = 80 × (72/80) × (72/80) = 80 × (9/10)² = 64.8 litres.
✗ Flipping the diagonals so the ratio comes out reversed | ✓ Assigning each difference to the opposite ingredient, then sense-checking
अलिगेशन: सस्ता : महँगा = (महँगा − माध्य) : (माध्य − सस्ता)। उत्तर सदा अनुपात होता है। प्रतिस्थापन में शेष शुद्ध मात्रा = आरंभिक × (1 − निकाला/कुल)^n।
Exam Pointer — SBI Clerk asks 1–2 mixture or alligation questions per shift, and alligation also speeds up average and partnership sums. Expect a two-item mix or a repeated-replacement type. Use the cross method for the ratio and the power formula for replacement; the trap the paper sets is a flipped diagonal, so sense-check which ingredient is larger.
60-Second Recap
- Alligation gives the mixing ratio by cross subtraction.
- Ratio = (dearer − mean) : (mean − cheaper).
- The value nearer the mean has the larger quantity.
- Replacement: remaining = initial × (1 − r/total)^n.