Average, Mixture & Alligation
Concept
Average is the most intuitive statistic. Imagine 5 students score 60, 70, 80, 90, and 100. Their average is (60+70+80+90+100) ÷ 5 = 400 ÷ 5 = 80. Every student “effectively” scored 80 — it’s the fair-share number. The formula Average = Sum ÷ Count works for any group of numbers, whether they’re ages, prices, speeds, or anything else.
Weighted Average comes into play when groups have different sizes. If Group A (10 students) averages 60 and Group B (20 students) averages 80, you can’t just average 60 and 80 to get 70. The 20 students from Group B count twice as much. The combined average = (10×60 + 20×80) ÷ (10+20) = (600 + 1600) ÷ 30 = 2200 ÷ 30 = 73.3. This is the single most important formula in this topic.
Alligation solves a specific practical problem: if you have rice costing ₹30/kg and rice costing ₹50/kg, what ratio must you mix them to get a mean price of ₹40/kg? The alligation rule says:
- Difference from mean for cheaper: 40 − 30 = 10 parts cheap
- Difference from mean for expensive: 50 − 40 = 10 parts expensive
- Ratio = 10 : 10 = 1 : 1
When quantities are equal, the mean price is the simple average. When quantities differ, the alligation rule automatically accounts for the weight difference.
Key Formulas
| Formula | Use |
|---|---|
| Average = (Sum of values) / (Number of values) | Finding the central value |
| Sum = Average × Number of values | Reversing the average |
| Missing number = Avg × n − (sum of rest) | Finding a missing value in a group |
| Weighted Avg = (n₁×avg₁ + n₂×avg₂) / (n₁+n₂) | Combining two groups |
| Alligation ratio = (d − m) : (m − c) | Finding mixing ratio (d=dear, c=cheap, m=mean) |
Worked Example
Q: A merchant mixes 20 kg of tea costing ₹200/kg with 30 kg of tea costing ₹300/kg. What is the price of the mixture per kg?
Step 1: Total cost of cheap tea = 20 × 200 = ₹4000 Step 2: Total cost of expensive tea = 30 × 300 = ₹9000 Step 3: Total mixture = 20 + 30 = 50 kg Step 4: Price per kg = (4000 + 9000) ÷ 50 = 13000 ÷ 50 = ₹260/kg
Answer: ₹260 per kg
Common Errors
- Simply averaging two averages (60+80)/2 = 70 when groups have different sizes → use weighted average formula
- Mixing up which price goes on which side of the alligation diagram → cheap on left, dear on right, mean in the middle
- Forgetting to multiply by quantity in alligation problems → always check: is it ratio by weight, or ratio by value?
📐 Diagram Reference
A balance-scale diagram for alligation: d1 on left, d2 on right, mean price d in the middle — the inversely proportional weights hang from each side
Diagrams are generated per-topic using AI. Support for AI-generated educational diagrams coming soon.