Q.The average salary per head of the entire staff of a small factory including the supervisor and labours is ₹5750. The average salary per head of the supervisor is ₹20,000 and that of the labours is ₹5000. Find the number of labours in the factory if there are 4 supervisors.

Weighted Average — The Intuition First

Imagine you have two test scores: a quiz where you scored 90/100, and a final exam where you scored 70/100. If you just take the ordinary average, you get $(90+70)/2 = 80$. But what if the final exam is more important than the quiz? Maybe the final counts for 80% of your grade and the quiz for only 20%. Then your final grade should reflect that importance — it should be pulled much closer to 70 than to 90.

That's the core idea of a weighted average: not every data point contributes equally. Some matter more, some matter less. The "weight" tells you how much each value should count.

The Precise Statement

A weighted average of a set of numbers $x_1, x_2, \dots, x_n$ is computed by multiplying each number by its corresponding weight $w_1, w_2, \dots, w_n$, summing those products, and then dividing by the sum of the weights.

The weights are usually positive numbers. They don't have to add up to 1 — the division by the total weight takes care of that automatically. But often, for convenience, we choose weights that do sum to 1 (like percentages that add to 100%). In that case, the formula simplifies to just:

The Grade Example Worked Out

Let's apply it to the quiz and final exam:

• Quiz: $x_1 = 90$, weight $w_1 = 20$ (or 0.20)
• Final: $x_2 = 70$, weight $w_2 = 80$ (or 0.80)

Using the general formula (weights as 20 and 80):

Your weighted average is 74, not 80. The final exam's larger weight pulled the average down toward 70.

Using the simplified formula with weights that sum to 1 ($0.20 + 0.80 = 1$):

Same result.

Why This Matters

Weighted averages appear everywhere in exams and real life:

• Final grades — different assignments have different weightage
• Economics — the Consumer Price Index weights different goods by how much people spend on them
• Physics — center of mass is a weighted average of positions, weighted by mass
• Statistics — when data comes from groups of different sizes, you weight each group's average by its size

One More Example — Different Weights, Same Idea

Suppose you buy three items: 2 kg of rice at ₹40/kg, 3 kg of dal at ₹60/kg, and 1 kg of sugar at ₹50/kg. What is the average price per kg?

You cannot just average ₹40, ₹60, and ₹50 — you bought different quantities. The quantity is the weight.

The weighted average price is about ₹51.67 per kg.

Weighted average is simply: each value gets to vote, but not everyone's vote counts the same. The weight is the number of votes.