Mixture & Alligation Short Notes – Quick Revision

Mixture & Alligation – Quick Revision Notes

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1. Basic Concept

• Combine two or more ingredients to get a mixture of desired mean value.

• Alligation formula gives the ratio of quantities.

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2. Key Formula – Two Ingredients

$$\text{Quantity ratio}=\frac{\text{Higher Price − Mean}}{\text{Mean − Lower Price}}$$

• H = Higher price

• L = Lower price

• M = Mean / desired price

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Example

Mix sugar @ ₹60/kg & ₹40/kg → desired ₹50/kg
Ratio = 60−50 : 50−40 = 10:10 = 1:1

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3. Multiple Ingredients

• Difference from mean → inverse ratio → quantity proportion

• Example: ₹60, ₹50, ₹40 → Mean = 50 → Ratio = 1:0:1

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4. Weighted Average (Mixture)

$$\text{Mean Price}=\frac{\text{Sum of (Quantity × Price)}}{\text{Total Quantity}}$$$$\text{Mean Price} = \frac{\sum (Quantity × Price)}{\sum Quantity}$$

• Example: 10 kg @ 20, 15 kg @ 30 → Mean = (200+450)/25 = 26/kg

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5. Profit & Loss in Mixtures

• Profit %:

$$\text{Profit \%}=\frac{\text{Selling Price (SP) − Cost Price (CP)}}{\text{CP}}\times 100$$

• Loss %:

$$\text{Loss \%}=\frac{\text{CP − SP}}{\text{CP}}\times 100$$

• Shortcut using Alligation: Treat SP as H and CP as L; use same alligation method to find ratio of ingredients if mixture involves profit/loss

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6. Replacing / Removing Mixture:

$$\text{New concentration} = \frac{\text{Remaining × Old \% + Added × New \%}}{\text{Total Quantity}}$$

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7. Quick Formulas Table

Concept	Formula
Two ingredients	(H−M):(M−L)
Mean price	Σ(Q×P)/ΣQ
Profit %	(SP−CP)/CP×100
Loss %	(CP−SP)/CP×100
Replacement	New % = (Remaining×Old% + Added×New%)/Total

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8. Practice Examples

1. Sugar 30 & 50 → desired 40 → Ratio = 1:1

2. 10 kg @ 20, 15 kg @ 30 → Mean = 26/kg

3. Milk 25 L, 5 L removed → New % = 80%

4. Acid 20% & 50% → 35% → Ratio = 1:1, total 30 L → 15 L each

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✅ One-Line Summary

“Mix quantities inversely proportional to difference from mean; weighted average gives mean price.”