SSC CGL Maths · Averages

Members Joining and Leaving

When one member joins or leaves, the change in the total tells you that member's value, and the change in the total is the count times the change in the average.

All Averages notes

Why this matters

Twenty questions in the bank, all EASY and sixteen of them from 2025: a new innings, a new student, a supervisor added, one number removed. The full method takes four lines; the fast route below takes one, and that saving matters at about 36 seconds a question.

Concept 1 of 2: A new member joins the group

The newcomer has to bring the old average for themselves, plus enough extra to lift every member of the new group, themselves included, by the rise. If the average falls, the newcomer brought less than the old average.

Definition

Full method: newcomer $=$ new total $-$ old total $= (n + 1)A' - nA$ , where $n$ members had average $A$ and the $n + 1$ members now have average $A'$ .
Fast route: newcomer $= A + (n + 1) \times$ rise. The rise is $A' - A$ ; it is negative if the average falls.
The same value from the new average: newcomer $= A' + n \times$ rise. Both forms are right; mixing them is the usual slip.
New average after a known newcomer $x$ : $A' = A + \frac{x - A}{n + 1}$ . The newcomer's distance from the old average is shared by all $n + 1$ .
Unknown old count: the newcomer's excess over the new average equals $n \times$ rise, so $n = \frac{x - A'}{A' - A}$ .

A member joins

x = A + (n + 1)(A' - A) \qquad A' = A + \frac{x - A}{n + 1}

nnumber of members before the newcomer joins
Aaverage before
A'average after
xthe newcomer's value

Worked example

A batsman's average over 14 innings is 36 runs. How many runs must he score in the 15th innings to raise his average to 38?

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 · Q58Easy

Example 1 · Averages · Members Joining and Leaving

The average age of 40 students in a class is 16 years. When a new student is admitted, the average increases by 0.2 years. What is the age of the new student?

Multiplying the rise by the wrong count

Starting from the old average, the newcomer must lift all

n + 1

members, so the extra is

(n + 1) \times

rise. Starting from the new average, it is

n \times

rise. Using

n

with the old average is short by exactly one rise.

Taking the new average as the newcomer's value

If the average rises after someone joins, the newcomer is above the new average, not equal to it. Only when the average does not change is the newcomer equal to it.

Drill 7 more on a new member joins the group

Concept 2 of 2: A member leaves the group

If the average falls when someone leaves, the leaver was above it and took more than their share away. Every member who stays is down by the fall, and the leaver carried that total away on top of the old average.

Definition

Full method: leaver $=$ old total $-$ new total $= nA - (n - 1)A'$ .
Fast route: leaver $= A + (n - 1) \times$ fall, where the fall is $A - A'$ . Only the $n - 1$ who stay carry the new average, so multiply by $n - 1$ . From the new average instead: leaver $= A' + n \times$ fall.
If the average rises when someone leaves, the fall is negative and the leaver was below the old average.
Several leave: the leavers' total $= nA - (n - k)A'$ , and their average is that total $\div k$ .
Average of those who stay after known values are removed: (old total $-$ removed values) $\div$ the number left. Removing values that average exactly $A$ leaves the average at $A$ .

A member leaves

x = A + (n - 1)(A - A') \qquad \text{Leavers' total} = nA - (n - k)A'

nnumber of members before anyone leaves
knumber who leave
Aaverage before
A'average of those who stay

Worked example

The average of 16 numbers is 30. One number is removed and the average of the rest is 29. Find the number removed.

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 · Q63Easy

Example 2 · Averages · Members Joining and Leaving

The average of 12 results is 75. If the top two results are excluded, the average becomes 72. What is the average of the top two results?

Multiplying the fall by the old count

When someone leaves, only the members who stay carry the new average. From the old average, the leaver is old average plus

(n - 1) \times

fall, not

n \times

fall. The

n \times

fall form belongs with the new average.

Thinking a fall means the leaver was low

If the average falls after someone leaves, the leaver was above the old average: removing a high value pulls the average down.

Drill 11 more on a member leaves the group

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 (2)

A new member joins the group
A member joins
$x = A + (n + 1)(A' - A) \qquad A' = A + \frac{x - A}{n + 1}$
A member leaves the group
A member leaves
$x = A + (n - 1)(A - A') \qquad \text{Leavers' total} = nA - (n - k)A'$

Watch out for (4)

Multiplying the rise by the wrong count
→ A new member joins the group
Taking the new average as the newcomer's value
→ A new member joins the group
Multiplying the fall by the old count
→ A member leaves the group
Thinking a fall means the leaver was low
→ A member leaves the group

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 20 questions