AFCAT Numerical Ability · Average

One Value In, Out or Changed

Turn the old average into a total, add or take away the value that changed, and divide by the right count: for a newcomer, someone who leaves, a member who is replaced and a value that was misread.

All Average notes

Why this matters

Six of the chapter's thirteen AFCAT past-year questions are on this page, from 2015 to 2019: a man replaced, a misread weight, a team's ages. Almost all of the page is EASY. The marks go when the count is not updated, or when a misread value is corrected in the wrong direction.

Concept 1 of 5: One more value: a new day, a teacher, a manager

A new value joins the total and the count grows by one. If the new value is above the average it pulls the average up, but by much less than the gap.

Definition

New total = old total + new value; new count = $n + 1$ .
Shortcut: the average moves by $\dfrac{x - A}{n + 1}$ , up if the new value $x$ is above the old average $A$ , down if below.
'Including the teacher' or 'the next day' is the same move.

One value added

\text{new average} = \dfrac{nA + x}{n + 1}

Worked example

A worker's average output over 9 days is 40 units. On the 10th day he makes 70. Find his new average.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Average · Adding, Removing or Replacing a Value

The average temperature of a city for

6

days was

28°C

. If the temperature on the seventh day was

35°C

, what is the average temperature for all

7

days?

The count goes up too

A salary of ₹60,000 added to 9 people averaging ₹40,000 gives

(3{,}60{,}000 + 60{,}000) \div 10 = ₹42{,}000

. Dividing by 9 again gives ₹46,667.

Drill 29 more on one more value: a new day, a teacher, a manager

Concept 2 of 5: One value removed

Take the value out of the total and one off the count. Removing a value above the average pulls the average down; removing one below it pushes the average up.

Definition

New total = $nA - x$ ; new count = $n - 1$ .
The average falls by $\dfrac{x - A}{n - 1}$ when $x$ is above the average, and rises when $x$ is below it.
Two or three values removed: take out their sum and reduce the count by that many.

One value removed

\text{new average} = \dfrac{nA - x}{n - 1}

Worked example

The average of 10 numbers is 24. The number 42 is removed. Find the new average.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Average · Adding, Removing or Replacing a Value

Six friends have an average height of 167 cm. A boy with height 162 cm leaves the group. Find the average height of the group now.

A low value leaving raises the average

From 11 values averaging 50, removing a 30 lifts the average to

(550 - 30) \div 10 = 52

Drill 8 more on one value removed

Concept 3 of 5: Finding the value that joined or left

If one newcomer lifts the average of

n

people by

d

, the newcomer must supply the new average for themselves plus

d

for each of the

n

others.

Definition

Joiner: $x = (n + 1) \times \text{new average} - n \times \text{old average}$ . Shortcut: $x = \text{new average} + n \times \text{rise}$ .
Leaver: $x = n \times \text{old average} - (n - 1) \times \text{new average}$ . Shortcut: $x = \text{old average} + (n - 1) \times \text{fall}$ .
If the average falls when someone joins, the 'rise' is negative and the newcomer is below the new average.

The newcomer

x = (n + 1)A' - nA = A' + n(A' - A)

Worked example

The average age of 19 students is 12. When the teacher is included, the average becomes 13.5. Find the teacher's age.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Average · Adding, Removing or Replacing a Value

The average weight of 21 boys was recorded as 64 kg. If the weight of the teacher was added, the average increased by 1 kg. What was the teacher's weight?

The newcomer pays for everyone

If 20 people average 50 and a newcomer lifts it to 52, the newcomer is

52 + 20 \times 2 = 92

, not

50 + 20 \times 2 = 90

Drill 13 more on finding the value that joined or left

Concept 4 of 5: Replacing a member

A replacement keeps the count the same, so the whole change in the total is spread over all

n

members. A rise of

d

in the average means the newcomer is

nd

more than the person replaced.

Definition

Count unchanged: new value − old value $= n \times$ change in the average.
Two replaced by two: the two newcomers' total is the two leavers' total plus $n \times$ the change; halve it for their average.
Ages in years and months: turn everything into months first.

One member replaced

x_{\text{new}} = x_{\text{old}} + n\,d

Worked example

The average weight of 8 boys rises by 1.5 kg when a boy of 46 kg is replaced by a new boy. Find the new boy's weight.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 02-214Easy

Example 4 · Average · Adding, Removing or Replacing a Value

The average weight of 5 men is increased by 2 kg when one of the men whose weight is 60 kg is replaced by a new man. The weight of the new man is

No one was added

A swap changes no count. A rise of 1.5 in the average of 8 means the total rose by

8 \times 1.5 = 12

, not

9 \times 1.5 = 13.5

Drill 18 more on replacing a member

Concept 5 of 5: A value misread or miscopied

The wrong value sat in the total in place of the right one. Swap it back: the total moves by the difference, and the average by the difference over the count.

Definition

Correct total = wrong total − wrong value + right value.
The average changes by $\dfrac{\text{right} - \text{wrong}}{n}$ : up if the right value is bigger, down if smaller.
Several errors: add up the differences first, then divide once.
Reversed digits: 53 written as 35 is an error of 18. Any two-digit reversal is off by 9 times the difference of the digits.

Correcting one value

\text{correct average} = A + \dfrac{\text{right value} - \text{wrong value}}{n}

Worked example

The average of 40 numbers was found to be 52. Later, 64 was found to have been copied as 40. Find the correct average.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 02-204Easy

Example 5 · Average · Adding, Removing or Replacing a Value

The average weight of a group of 20 boys was calculated to be 89.4 kg and it was later discovered that one weight was misread as 78 kg instead of 87 kg. The correct average weight is

Which way round

If 64 was written as 46, the total was 18 too small and the true average is higher. Subtracting instead moves the average the wrong way by twice as much.

Drill 20 more on a value misread or miscopied

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 (5)

One more value: a new day, a teacher, a manager
One value added
$\text{new average} = \dfrac{nA + x}{n + 1}$
One value removed
One value removed
$\text{new average} = \dfrac{nA - x}{n - 1}$
Finding the value that joined or left
The newcomer
$x = (n + 1)A' - nA = A' + n(A' - A)$
Replacing a member
One member replaced
$x_{\text{new}} = x_{\text{old}} + n\,d$
A value misread or miscopied
Correcting one value
$\text{correct average} = A + \dfrac{\text{right value} - \text{wrong value}}{n}$

Watch out for (5)

The count goes up too
→ One more value: a new day, a teacher, a manager
A low value leaving raises the average
→ One value removed
The newcomer pays for everyone
→ Finding the value that joined or left
No one was added
→ Replacing a member
Which way round
→ A value misread or miscopied

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