Mixture and Alligation ke questions competitive exams mein bahut common hote hain. Yeh topic profit-loss aur ratio ke concepts se linked hota hai. Neeche diye gaye sabhi formulas, theorems aur tricks aapke blog ke liye SEO-friendly format mein ready hain.
📘 Basic Concepts
- Mixture: Jab do ya zyada components (jaise milk + water) ko mix kiya jata hai.
- Alligation Rule: Ek shortcut method hai average value nikalne ke liye jab do alag-alag values mix ho rahi ho.
| No. | 📘 Theorem Name | 🧠 Concept Summary (Hinglish) |
|---|---|---|
| 1 | Alligation Rule | Cheaper aur dearer values ka ratio nikalne ka shortcut |
| 2 | Weighted Average | Multiple items ka average value calculate karna |
| 3 | Equal Quantity Shortcut | Jab quantity same ho, simple average use hota hai |
| 4 | Successive Replacement | Liquid replace hone par final quantity ka formula |
| 5 | Alligation Line Method | Visual method for ratio calculation |
| 6 | Mixture Purity | Mixture mein pure part ka percentage |
| 7 | Profit-Based Alligation | Cost aur selling price ka mix for profit calculation |
| 8 | Multiple Mixtures | 3+ items mix karne ka stepwise method |
| 9 | Time-Based Mixing | Continuous inflow-outflow ka net effect |
| 10 | Reverse Alligation | Given average se ratio nikalna |
🔹 1. Alligation Rule
📌 Formula: (Cheaper – Mean) : (Mean – Dearer) = Quantity Ratio
Examples:
- Milk ₹20/litre + ₹30/litre → Mixture ₹25/litre ⇒ Ratio = (30–25):(25–20) = 5:5 = 1:1
- Alcohol 10% + Alcohol 40% → Mixture 25% ⇒ Ratio = (40–25):(25–10) = 15:15 = 1:1
- Sugar ₹15/kg + ₹25/kg → Mixture ₹20/kg ⇒ Ratio = (25–20):(20–15) = 5:5 = 1:1
🔹 2. Weighted Average Theorem
📌 Formula: Average = (Q₁×V₁ + Q₂×V₂ + … + Qₙ×Vₙ) / (Q₁ + Q₂ + … + Qₙ)
Where:
- Q = Quantity
- V = Value (price, concentration, etc.)
Examples:
- 2L milk @ ₹20 + 3L milk @ ₹30 ⇒ Average = (2×20 + 3×30) / 5 = ₹26
- 5kg rice @ ₹40 + 7kg rice @ ₹50 ⇒ Average = (5×40 + 7×50)/12 = ₹46.67
- 3L alcohol @ 10% + 2L @ 30% ⇒ Average = (3×10 + 2×30)/5 = 18%
🔹 3. Equal Quantity Shortcut ( if both quantity equal )
📌 Formula: Average = (Value₁ + Value₂) / 2
Examples:
- ₹20 + ₹30 milk = ₹25/litre
- 10% + 40% alcohol = 25%
- ₹15 + ₹25 sugar = ₹20/kg
🔹 4. Successive Replacement Theorem
📌 Formula: Final Quantity = Initial × (1 – Replaced/Total)ⁿ
Where:
- n = Number of replacements
Examples:
- 20L milk, 5L replaced once ⇒ Final = 20 × (1 – 5/20) = 15L
- 10L alcohol, 2L replaced twice ⇒ Final = 10 × (1 – 2/10)² = 6.4L
- 30L solution, 6L replaced thrice ⇒ Final = 30 × (1 – 6/30)³ = 17.576L
🔹 5. Alligation Line Method
📌 Formula: Draw line → Left: Cheaper | Middle: Mean | Right: Dearer Then subtract crosswise to get ratio.
Ek line draw karo:
- Left end pe cheaper value
- Right end pe dearer value
- Middle mein mean value
- Cross subtraction se ratio milta hai
🧪 Examples:
- ₹20 & ₹30 → ₹25 ⇒ Ratio = (30–25):(25–20) = 5:5 = 1:1
- 10% & 40% → 25% ⇒ Ratio = (40–25):(25–10) = 15:15 = 1:1
- ₹15 & ₹25 → ₹20 ⇒ Ratio = (25–20):(20–15) = 5:5 = 1:1
🔹 6. Mixture Purity Theorem
📌 Formula: Purity % = (Pure Quantity / Total Quantity) × 100
Agar ek mixture mein kuch part impure ho (jaise alcohol-water) toh
🧪 Examples:
- 8L milk + 2L water ⇒ Purity = (8/10)×100 = 80%
- 6L alcohol + 4L water ⇒ Purity = (6/10)×100 = 60%
- 9L petrol + 1L kerosene ⇒ Purity = (9/10)×100 = 90%
🔹 7. Profit-Based Alligation
📌 Formula: Profit % = (Selling Price – Cost Price) / Cost Price × 100
Jab cost price aur selling price ka mixture ho toh use this with alligation to find best mix for max profit.
Examples:
- Cost ₹20, Sell ₹25 ⇒ Profit = (25–20)/20 × 100 = 25%
- Mix ₹15 & ₹30 → Sell @ ₹25 ⇒ Use alligation to find ratio and profit
- ₹10 & ₹40 mix → Sell @ ₹30 ⇒ Apply alligation + profit formula
🔹 8. Multiple Mixtures Theorem
📌 Formula: Stepwise alligation: Mix first two → then mix result with third
Jab 3 ya zyada items mix ho rahe ho:
Step-by-step pairwise alligation apply karo → Pehle 2 ka average lo → Usko teesre ke saath mix karo
🧪 Examples:
- ₹20 & ₹30 → ₹25, then mix with ₹40 ⇒ Step 1: ₹25, Step 2: ₹25 & ₹40 → ₹32.5
- Alcohol 10%, 30%, 50% ⇒ Step 1: 10 & 30 → 20%, Step 2: 20 & 50 → 35%
- Rice ₹40, ₹50, ₹60 ⇒ Step 1: ₹40 & ₹50 → ₹45, Step 2: ₹45 & ₹60 → ₹52.5
🔹 9. Time-Based Mixing Theorem
📌 Formula: Final Quantity = Initial × (1 – Replaced/Total)ⁿ
Agar ek tank continuously fill ho raha ho aur simultaneously kuch liquid nikal raha ho:
Use differential rate concept → Net rate = Inflow – Outflow → Apply replacement formula if needed
Examples:
- Tank 100L, 10L replaced daily for 3 days ⇒ Final = 100 × (1 – 10/100)³ = 72.9L
- 50L milk, 5L replaced twice ⇒ Final = 50 × (1 – 5/50)² = 40.5L
- 30L solution, 6L replaced once ⇒ Final = 30 × (1 – 6/30) = 24L
🔹 10. Reverse Alligation Theorem
📌 Formula: Use basic alligation in reverse to find ratio
Agar mixture ka average diya ho aur ratio nikalna ho:
Use basic alligation rule in reverse → Mean value se dono ends subtract karo → Ratio milta hai
🧪 Examples:
- Mixture ₹25, items ₹20 & ₹30 ⇒ Ratio = (30–25):(25–20) = 1:1
- Mixture 25% alcohol, items 10% & 40% ⇒ Ratio = (40–25):(25–10) = 1:1
- Mixture ₹20, items ₹15 & ₹25 ⇒ Ratio = (25–20):(20–15) = 1:1
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