SSC CGL Maths · Averages

Sum and Mean

The average is the total divided by the number of items, so the total is always the number of items times the average.

Why this matters

Fourteen questions in the bank, eleven EASY and three MODERATE, seven of them from 2025. Every later page in this chapter rests on one habit taught here: change each average into a total, work with totals, and divide only at the end.

Concept 1 of 3: Average, sum and count

Pool everything together and share it out equally: the equal share each item gets is the average. So the average and the total carry the same information, and you can move from one to the other in one step.

Definition

Average $=$ sum $\div$ count.
Turned round: sum $=$ count $\times$ average. Most SSC averages questions are solved by changing every average into a sum, adding or subtracting the sums, and dividing once at the end.
The count is the number of items. Sometimes you must find it first, for example the number of items bought with the money that is left.
When a value is described by its link to another ( $20\%$ more than the second), write every value in terms of one unknown, add them, and set the result equal to the sum.
When a question gives limits for a whole-number value, keep only the values that every condition allows, then average those values.

Average, sum and count

\text{Average} = \frac{\text{Sum}}{\text{Count}} \qquad \text{Sum} = \text{Count} \times \text{Average}

Worked example

Four friends share a taxi. Three of them pay ₹120 each and the fourth pays ₹160. What is the average amount paid?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 24 Sep 2025, Shift 3 · Q68Easy

Example 1 · Averages · Sum and Mean

The average of three numbers is 35. The first number is

20\%

more than the second, and the third is

20\%

less than the second. What is the second number?

Averaging two averages of unequal groups

If 4 numbers average 10 and 6 other numbers average 20, the average of all 10 is not 15. Change both to sums first:

\frac{40 + 120}{10} = 16

Forgetting part of the count

In a mixed purchase the average price is total money over total items. Count every item, including the ones whose number you had to work out from the money left over.

Drill 7 more on average, sum and count

Concept 2 of 3: Average from an assumed value

Adding five three-digit numbers is slow. Pick a round number near the middle, note how far each value is above or below it, and average those small gaps instead. The true average is your round number plus that small correction.

Definition

Choose an assumed average $A$ , a round number close to the values.
Write each deviation $x - A$ : positive for a value above $A$ , negative for a value below.
True average $= A +$ (sum of deviations) $\div$ count.
Deviations from the true average always add to 0. That gives a quick check, and later in this chapter it gives whole methods.
The shortcut is always safe: it is the sum rule done with smaller numbers.

Average from an assumed value

\bar{x} = A + \frac{\sum (x - A)}{n} \qquad \sum (x - \bar{x}) = 0

Athe assumed average, any convenient number
nthe number of values
$\bar{x}$ the true average

Worked example

Find the average of 87, 92, 85, 96 and 93.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 19 Sep 2024, Shift 1 · Q62Easy

Example 2 · Averages · Sum and Mean

A student studying in 6th class scored the following marks in 6 subjects. The maximum marks for each subject are 100.

Subjects	Marks
Hindi	98
English	97
Mathematics	93
Social Science	94
Computer	92
Science	93

What are the average marks scored by the student?

Dropping the sign of a deviation

A value below the assumed average has a negative deviation. Adding every gap as positive pushes the average too high.

Adding the deviation total without dividing

The sum of deviations must be divided by the count before it is added to the assumed average. Adding the whole total overshoots by a factor of the count.

Drill 2 more on average from an assumed value

Concept 3 of 3: Pair averages: add all the statements

When each statement gives the total of a few people, adding all the statements counts every person the same number of times. Divide by that number and you have the grand total in one step, without solving for anyone first.

Definition

Change each average into a sum: 'the average of F and M is 5,000' means $F + M = 10000$ .
If every pair of three people is given, adding the three pair sums counts each person twice, so the total of all three is half of that.
One person $=$ the total of all three minus the pair that leaves that person out.
For other symmetric statements, count how many times each unknown appears when all the statements are added. With sum $S$ , 'the average of two of them plus the third, $x$ ' is $\frac{S - x}{2} + x = \frac{S + x}{2}$ , and adding all three such statements gives $\frac{3S + S}{2} = 2S$ .

Total from three pair sums

a + b + c = \frac{(a + b) + (b + c) + (c + a)}{2}

Worked example

The average age of A and B is 20 years, of B and C is 25 years, and of A and C is 27 years. What is B's age?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 1 · Q57Moderate

Example 3 · Averages · Sum and Mean

Three numbers are such that when the average of any two of them is added to the third, the results obtained are 180, 168, and 150, respectively. What is the average of the original three numbers?

Forgetting to halve the total of the pair sums

When all three pair sums are added, each person appears in two of them, so the result is twice the total of the three people.

Reading an average as a sum

'The average of F and M is 5,000' means F and M together are 10,000. Using 5,000 as their sum halves every answer that follows.

Drill 2 more on pair averages: add all the statements

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 (3)

Average, sum and count
Average, sum and count
$\text{Average} = \frac{\text{Sum}}{\text{Count}} \qquad \text{Sum} = \text{Count} \times \text{Average}$
Average from an assumed value
Average from an assumed value
$\bar{x} = A + \frac{\sum (x - A)}{n} \qquad \sum (x - \bar{x}) = 0$
Pair averages: add all the statements
Total from three pair sums
$a + b + c = \frac{(a + b) + (b + c) + (c + a)}{2}$

Watch out for (6)

Averaging two averages of unequal groups
→ Average, sum and count
Forgetting part of the count
→ Average, sum and count
Dropping the sign of a deviation
→ Average from an assumed value
Adding the deviation total without dividing
→ Average from an assumed value
Forgetting to halve the total of the pair sums
→ Pair averages: add all the statements
Reading an average as a sum
→ Pair averages: add all the statements

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 14 questions