Averages · Topic 5 of 6

⚖️ Weighted & Special Sets

2 exam question types, fully solved

When two groups have different sizes — 40 boys and 60 girls, skilled and unskilled workers — you cannot just average their averages. The bigger group counts for more. This is the weighted average, and it is where most marks are quietly lost.

Totals on top, counts on the bottom

Weight each group's average by how many people it has, then divide by the total head-count:

The number-one trap: is only correct when the groups are the same size. Otherwise the answer always leans towards the average of the larger group. If 60 students score 45 and 40 score 55, the answer must dip below the midpoint 50 — because the 45-group is bigger.

Finding a missing group average: take the grand total, subtract the known group's total, then divide by the missing group's own count — not the overall count.

Speed tip: shrink the counts to a simple ratio first. 40 : 60 becomes 2 : 3, so the answer splits the gap in a 3 : 2 ratio leaning towards the bigger side.

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Weighted Average = (Σ value × weight) ÷ Σ weight. Use when items have different importance (marks with credits, profit with different category weights, etc.)

ValueWeight

Numerator: (80×3) + (60×2) + (90×5) = 810

Denominator: 3 + 2 + 5 = 10

Weighted Avg = 810 ÷ 10 = 81

Sections of 40 (avg 55) and 60 (avg 45): combined .

Q1School P: 200 students average 75; School Q: 300 students average 65. The combined average is:

Q2A school has 60 boys (average weight 50 kg) and 40 girls (average 45 kg). The average weight of all students is:

Q3Department X has 15 employees averaging ₹8,000; Department Y has 25 averaging ₹6,400. The combined average is:

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Real exam questions — Weighted & Special Sets

2 question types · 8 solved examples from real SSC papers

Combined average = total of totals ÷ total of counts. Averaging the averages is wrong unless the groups are the same size.

How to solve this type

The combined average of groups is

\dfrac{n_1 A_1 + n_2 A_2 + \dots}{n_1 + n_2 + \dots}

— totals on top, counts on the bottom.
The single biggest trap in all of averages:

\dfrac{A_1 + A_2}{2}

is WRONG unless the groups are the same size. The combined answer always leans towards the average of the larger group.
To find a missing group average: take the grand total, subtract the known group's total, and divide by the missing group's own count.

Two sections of 40 and 60 students have averages 55 and 45. Find the overall average.

A52B51C50D49

200 workers include 80 skilled (average wage ₹400) and 120 unskilled. The combined average is ₹300. Find the unskilled average.

A₹240B₹233C₹220D₹250

A company has 20 managers (average salary ₹50,000) and 80 staff (average ₹20,000). What is the overall average salary?

A₹26,000B₹28,000C₹30,000D₹27,000

In an exam there are 3 Maths papers averaging 45, 2 English averaging 55, and 2 Hindi averaging 60. Find the overall average per paper.

A52.1B56.5C54.1D55.6