AFCAT Numerical Ability · Average

Groups That Join or Leave

When a whole group with its own average joins or leaves, add or take away its total and change the count by its size. When some leave and others join, do both before you divide.

All Average notes

Why this matters

This is the chapter's biggest page and nearly a quarter of it is HARD. Most of the HARD questions have some people leaving and others joining at once. The slip is to change the total but not the count, or to change the count for only one of the two groups.

Concept 1 of 4: A group joins

A group of

m

with average

B

brings a total of

mB

. Add it to the old total, add

m

to the count, and divide.

Definition

New total = $nA + mB$ (or the sum of the listed values); new count = $n + m$ .
The new average lies between $A$ and $B$ , nearer the bigger group.
Shortcut: the average moves by $\dfrac{m(B - A)}{n + m}$ .

A group joins

\bar x' = \dfrac{nA + mB}{n + m}

Worked example

The average height of 24 players is 170 cm. Six players averaging 180 cm join. Find the new average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Average · Members Joining or Leaving

The average age of

15

students in a class is

22

years. If

5

new students with an average age of

18

years join the class, what will be the new average age of all students?

The size of the joining group matters

Six people averaging 80 joining 24 who average 70 lift the average to

(1680 + 480) \div 30 = 72

, not to the midpoint 75.

Drill 28 more on a group joins

Concept 2 of 4: A group leaves

Take the leavers' total out of the old total and their number off the count. If those who leave were above average, the rest are below it.

Definition

New total = $nA - mB$ ; new count = $n - m$ .
If the leavers' values are listed one by one, add them to get $mB$ .
If the leavers average exactly $A$ , the average of the rest is still $A$ .

A group leaves

\bar x' = \dfrac{nA - mB}{n - m}

Worked example

The average salary of 30 employees is ₹40,000. Six employees averaging ₹52,000 leave. Find the new average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Average · Members Joining or Leaving

In a group of

20

people, the average weight is

65

kg. If

4

people with an average weight of

70

kg leave the group, what is the new average weight?

Leaving can raise the average

If 6 of 30 students averaging 50 leave and those 6 averaged 38, the rest average

(1500 - 228) \div 24 = 53

, above the old 50.

Drill 35 more on a group leaves

Concept 3 of 4: Some leave, others join

Do the leaving and the joining on the total, do the same on the count, and divide once at the end.

Definition

New total = old total − leavers' total + joiners' total.
New count = $n - m + k$ . When as many join as leave (a group replaced), the count does not change.
Write four numbers in a line before you calculate: old total, out, in, new count.

Leave and join together

\bar x' = \dfrac{nA - mB + kC}{n - m + k}

Worked example

A class of 32 averages 60. Nine students averaging 45 leave and seven averaging 75 join. Find the new average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Average · Members Joining or Leaving

In a class of

28

students, the average marks in English is

72

. If

8

students with an average of

65

marks leave and

4

students with an average of

88

marks join, what is the new average?

Two changes to the count

If 9 leave and 7 join a class of 32, the new count is 30. Using

32 - 9 = 23

32 + 7 = 39

gives a wrong average.

Drill 19 more on some leave, others join

Concept 4 of 4: Finding the group that joined or left

Work backwards from the two averages. The gap between the new total and the old total is what the group brought in or took away.

Definition

Joined: group total = new total − old total.
Left: group total = old total − new total.
Shortcut for joiners: $B = A' + \dfrac{n(A' - A)}{m}$ .
A group average of 0 (brand-new books, newborn babies) is possible: the total did not change at all.

The joining group's total

mB = (n + m)A' - nA

Worked example

The average of 40 numbers is 25. Ten more numbers are added and the average becomes 28. Find the average of the new numbers.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Average · Members Joining or Leaving

The average score of a class of 45 students was 68. When 5 new students joined, the average increased by 2. What was the average score of the new students?

Divide by the group, not by everyone

If ten joiners add 400 to the total, they average

400 \div 10 = 40

each, not

400 \div 50

Drill 8 more on finding the group that joined or left

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)

A group joins
A group joins
$\bar x' = \dfrac{nA + mB}{n + m}$
A group leaves
A group leaves
$\bar x' = \dfrac{nA - mB}{n - m}$
Some leave, others join
Leave and join together
$\bar x' = \dfrac{nA - mB + kC}{n - m + k}$
Finding the group that joined or left
The joining group's total
$mB = (n + m)A' - nA$

Watch out for (4)

The size of the joining group matters
→ A group joins
Leaving can raise the average
→ A group leaves
Two changes to the count
→ Some leave, others join
Divide by the group, not by everyone
→ Finding the group that joined or left

Drill the questions on this topic

94 questions from the bank, with cart and Word export.

Drill all 94 questions