SSC CGL Maths · Averages

Weighted and Combined Averages

When groups of different sizes are pooled, the combined average is the total of all groups over the total count, so each group's average counts in proportion to its size.

All Averages notes

Why this matters

Twenty-seven questions in the bank, the biggest page in this chapter and all EASY. Eight questions from the 2024 shifts share one shape: 30 students, of whom 12 have one average and the other 18 another. The full method is safe; the fast route and the options often finish it in one line.

Concept 1 of 4: Combined average of two groups

Pool the groups: add their totals, add their counts, and divide. The result always lies between the two group averages, and it sits nearer the average of the bigger group, because that group has more members pulling on it.

Definition

Full method: combined $= \frac{n_1A_1 + n_2A_2}{n_1 + n_2}$ .
Fast route: start at one group's average and move towards the other's by the other group's share of the count: $A_1 + \frac{n_2}{n_1 + n_2}(A_2 - A_1)$ .
It equals the plain average of $A_1$ and $A_2$ only when the groups are the same size. If both groups have the same average, the combined average is that same value.
Option check: the answer lies between the two averages and nearer the larger group. That often removes two options before any working.

Combined average

\bar{x} = \frac{n_1A_1 + n_2A_2}{n_1 + n_2} \qquad \bar{x} = A_1 + \frac{n_2}{n_1 + n_2}(A_2 - A_1)

n_1, n_2the sizes of the two groups
A_1, A_2their averages

Worked example

In a test, 15 boys average 54 marks and 25 girls average 62 marks. What is the average of all 40 students?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 1 · Q68Easy

Example 1 · Averages · Weighted and Combined Averages

In a company, the average salary of 5 junior employees is Rs. 30,000, and that of 4 senior employees is Rs. 60,000. What is the combined average salary?

Averaging the two averages

Groups of 3 and 7 averaging 40 and 80 do not average 60 together: weight by size,

\frac{120 + 560}{10} = 68

. The plain average is right only for equal groups.

Choosing an option outside the two averages

A combined average can never be below the smaller group average or above the larger one. An option outside that range is wrong at sight.

Drill 14 more on combined average of two groups

Concept 2 of 4: Weights given as percentages or a ratio

A weight is just a group's share of the whole. Percentages and ratios are counts in disguise, so the combined-average rule works unchanged.

Definition

Weights as percentages that add to $100\%$ : weighted average $= w_1x_1 + w_2x_2 + \cdots$ , with each weight written as a decimal.
Weights as a ratio $a : b : c$ : treat $a$ , $b$ , $c$ as the counts and divide by $a + b + c$ .
A loss or a negative return enters with its minus sign.
Fast route: take deviations from a convenient value, weight them, and add the result to that value.

Weighted average

\bar{x} = \frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1 + w_2 + w_3}

wa weight: a count, a ratio term or a percentage
xthe value carrying that weight

Worked example

A final grade counts homework 30%, a project 20% and an exam 50%. A student scores 80 in homework, 90 in the project and 70 in the exam. What is her final grade?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 2 · Q60Easy

Example 2 · Averages · Weighted and Combined Averages

A company has 80 employees, of whom

35\ \%

are women. The average monthly salary of the women is ₹72,000, while that of the men is ₹68,000. What is the overall average monthly salary (in ₹ 000) of the whole workforce?

Dropping the sign of a loss

In a weighted average a 5% loss enters as −5. Adding it as +5 overstates the result by twice its weighted share.

Ignoring the weights

Marks of 75, 80 and 60 with weights 40%, 50% and 10% do not average

\frac{75 + 80 + 60}{3}

. Each mark counts by its weight, and the light 60 barely moves the result.

Drill 3 more on weights given as percentages or a ratio

Concept 3 of 4: Finding a missing group's average

Run the combined average backwards. The total of everyone, minus the known group's total, is the other group's total. Or think of a balance: the known group's gap above the overall average is matched by the other group's gap below it.

Definition

Full method: missing total $= nA - n_1A_1$ ; missing average $=$ that total $\div (n - n_1)$ .
Fast route (balance): $n_1(A_1 - A) = n_2(A - A_2)$ . The known group's total excess over the average is the other group's total shortfall.
With three or more groups, take every known group's total away from the grand total, then divide what is left by the members left.

Missing group's average

A_2 = \frac{nA - n_1A_1}{n - n_1} \qquad n_1(A_1 - A) = n_2(A - A_2)

n, Asize and average of everyone
n_1, A_1the known group
n_2, A_2the missing group, with $n_2 = n - n_1$

Worked example

The average weight of 40 students is 50 kg. The 15 boys average 56 kg. What is the average weight of the girls?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 3 · Q65Easy

Example 3 · Averages · Weighted and Combined Averages

The average score of 25 students in a test is 68. The top 5 scorers had an average of 80, and the lowest 5 had an average of 50. What is the average of the remaining 15 students?

Dividing the missing total by the whole count

The total left after removing the known group belongs only to the remaining members, so divide by the number remaining, not by the size of the whole group.

Drill 3 more on finding a missing group's average

Concept 4 of 4: Alligation: group sizes from the averages

The combined average is a balance point. Each group's distance from it is in inverse proportion to its size: the bigger group sits closer to the combined average.

Definition

Rule of alligation: $n_1 : n_2 = (A_2 - A) : (A - A_1)$ , where the combined average $A$ lies between $A_1$ and $A_2$ .
Write $A_1$ and $A_2$ on top, $A$ in the middle, and take the differences across: each group gets the other group's distance.
Once the ratio is known, scale it to the count you are given.
The full method gives the same answer: write the combined-average equation and solve for the unknown count.
The rule is for two groups. With three groups, go back to totals.

Rule of alligation

\frac{n_1}{n_2} = \frac{A_2 - A}{A - A_1}

Athe combined average, between $A_1$ and $A_2$

Worked example

Boys in a class average 60 marks and girls average 72 marks. The class average is 65. Find the ratio of boys to girls.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 15 Sep 2025, Shift 3 · Q56Easy

Example 4 · Averages · Weighted and Combined Averages

The overall average salary for all employees in an office is ₹16,000. Officers earn an average salary of ₹28,000, while non-officers make an average of ₹10,000. If there are 15 officers, determine the number of non-officers.

Giving each group its own distance

In alligation each group's share is the OTHER group's distance from the combined average. A combined average close to group 1's average means group 1 is the bigger group.

Drill 3 more on alligation: group sizes from the averages

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Combined average of two groups
Combined average
$\bar{x} = \frac{n_1A_1 + n_2A_2}{n_1 + n_2} \qquad \bar{x} = A_1 + \frac{n_2}{n_1 + n_2}(A_2 - A_1)$
Weights given as percentages or a ratio
Weighted average
$\bar{x} = \frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1 + w_2 + w_3}$
Finding a missing group's average
Missing group's average
$A_2 = \frac{nA - n_1A_1}{n - n_1} \qquad n_1(A_1 - A) = n_2(A - A_2)$
Alligation: group sizes from the averages
Rule of alligation
$\frac{n_1}{n_2} = \frac{A_2 - A}{A - A_1}$

Watch out for (6)

Averaging the two averages
→ Combined average of two groups
Choosing an option outside the two averages
→ Combined average of two groups
Dropping the sign of a loss
→ Weights given as percentages or a ratio
Ignoring the weights
→ Weights given as percentages or a ratio
Dividing the missing total by the whole count
→ Finding a missing group's average
Giving each group its own distance
→ Alligation: group sizes from the averages

Test yourself on Averages

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Averages chapter test Drill all 27 questions