Weighted Average Mixture — First Principles
Imagine you have two buckets of sand. One bucket is fine sand, the other is coarse gravel. If you take one handful from each and mix them, the mixture is somewhere between fine and coarse — but exactly where depends on how big your handfuls were. If you grabbed a huge scoop of fine sand and just a pinch of gravel, the mixture will feel almost like fine sand. That's the core idea: a weighted average tells you the result of mixing things when the ingredients contribute unequally.
The Intuition: Why "Weighted"?
A simple average (arithmetic mean) treats every item equally. If you scored 80 and 90 on two tests, the average is 85 — each test gets equal say. But what if the first test was worth 20% of your grade and the second was worth 80%? Then your final grade is not 85. The second test matters more. You weight it more heavily.
In a weighted average mixture, each component has two properties:
- Its value (e.g., concentration, price, speed, percentage)
- Its weight (how much of it is in the mixture — mass, volume, number of items, etc.)
The mixture's final value is the sum of (value × weight) for each component, divided by the total weight.
Mixture value=w1+w2+⋯+wnw1v1+w2v2+⋯+wnvn
A Concrete Example
Suppose you mix two types of rice:
- Type A: ₹50 per kg, you take 3 kg
- Type B: ₹80 per kg, you take 2 kg
What is the price per kg of the mixture?
If you just averaged the prices: (50 + 80)/2 = ₹65. But that's wrong — you have more of the cheaper rice. The correct calculation:
Price=3+2(3×50)+(2×80)=5150+160=5310=62
The mixture costs ₹62 per kg. Notice it's closer to ₹50 (the cheaper rice) because there's more of it. The weights (3 kg and 2 kg) "pull" the average toward the component with larger weight.
The Precise Statement
A weighted average mixture is a convex combination of the component values, where the coefficients (weights) are non-negative and sum to 1 after normalisation. In plain language:
The mixture's value always lies between the smallest and largest component values. It equals a component's value only if all other components have zero weight. The more weight a component has, the more it pulls the mixture toward its own value.
Common Exam Contexts
You'll see weighted averages in:
- Alligation (mixing two things to get a target concentration)
- Profit/loss on mixtures of goods
- Speed when distances or times are unequal
- Averages of groups with different sizes (e.g., average height of boys and girls combined)
When solving mixture problems, always ask: What is the "value" being averaged, and what is the "weight" that determines its influence? The weight is usually mass, volume, quantity, or number of items — never the price or percentage itself.
A Common Mistake
Students often add the values and divide by the number of components, forgetting the weights. If you mix 1 litre of milk at 3% fat and 100 litres at 6% fat, the fat percentage is not 4.5%. It's overwhelmingly close to 6% because the second component dominates. Always check: does each component contribute equally? If not, you need a weighted average.