AFCAT Numerical Ability · Average

The Average, Its Total and the First n Numbers

An average is a total shared out equally: find it, run it backwards to a total or a missing value, see how it moves when every value changes, and use the shortcuts for evenly spaced numbers.

All Average notes

Why this matters

This page has only two HARD questions, but every later page leans on it: almost every average problem becomes a total, a subtraction and a division. Marks go on two slips, dividing by the wrong count and forgetting that adding the same number to every value moves the average by exactly that number.

Concept 1 of 5: Average, total and count

An average is what each value would be if the total were shared out equally. Know any two of the average, the total and the count, and you have the third.

Definition

Average = total ÷ count. So total = average × count, and count = total ÷ average.
Work with totals, not averages. Two averages cannot be added or subtracted directly; two totals can.
The average always lies between the smallest and the largest value. Use that to strike out options.

Arithmetic mean

\text{Average} = \dfrac{\text{Sum of the values}}{\text{Number of values}}

Worked example

Find the average of 14, 9, 23, 17 and 12.

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The same idea in a practice question:

Practice questionEasy

Example 1 · Average · Basic and Simple Averages

A student scores

85

92

78

96

, and

89

in five subjects. What is his average score?

An average of averages

Two classes averaging 60 and 70 do not average 65 unless they are the same size. Add the two totals, then divide by everyone.

Zero is a value

A score of 0 still counts in the count: 10, 0 and 20 average 10, not 15.

Drill 15 more on average, total and count

Concept 2 of 5: A missing value, or the score needed for a target

If you know the average you want, you know the total you need. The missing value is that total minus everything you already have.

Definition

Missing value = average × count − the sum of the known values.
'What must he score next to average $A$ over $n$ tests?' The needed total is $A \times n$ ; subtract the scores so far.
Deviation shortcut: add up how far each known value is above or below the target. The missing value must cancel that. For a target of 50 with 47, 52 and 55, the deviations are $-3, +2, +5$ , a total of $+4$ , so the fourth value is $50 - 4 = 46$ .

The missing value

x = A \times n - (\text{sum of the other } n - 1 \text{ values})

Worked example

The average of five numbers is 26. Four of them are 21, 30, 18 and 33. Find the fifth.

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The same idea in a practice question:

Practice questionEasy

Example 2 · Average · Basic and Simple Averages

A student scores

85

90

78

, and

92

in four tests. What should he score in the fifth test to have an average of

86

Count the test still to come

For a target average over 5 tests, the needed total is 5 times the target, not 4 times. The test you are about to sit is part of the count.

Drill 13 more on a missing value, or the score needed for a target

Concept 3 of 5: When every value changes

Add 5 to every value and the average rises by 5. Double every value and the average doubles. Change only some values, and the total moves by the sum of the changes.

Definition

Add or subtract $k$ from every value: the average moves by $k$ .
Multiply or divide every value by $k$ : the average is multiplied or divided by $k$ . Apply the steps in their order: × 3 and then − 5 turns $A$ into $3A - 5$ .
When only some values change, find the total change and divide by the full count. If 4 of 20 values each rise by 5, the total rises by 20 and the average by 1.
If the total must stay the same with fewer items, the new average is that total ÷ the new count.

A change applied to every value

x_i \to a\,x_i + b \;\Rightarrow\; \bar x \to a\,\bar x + b

Worked example

The average of 15 numbers is 30. Each number is multiplied by 4, and then 6 is subtracted from each. Find the new average.

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The same idea in a practice question:

Practice questionModerate

Example 3 · Average · Basic and Simple Averages

The average of

6

numbers is

25

. If each of the first

3

numbers is increased by

2

and each of the last

3

numbers is decreased by

4

, what will be the new average?

Order matters

Doubling and then adding 3 turns an average of 12 into 27. Adding 3 first and then doubling gives 30.

Some values, not all

If only 6 of 30 values rise by 5, the total rises by 30 and the average by 1, not by 5.

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Concept 4 of 5: Evenly spaced numbers: first plus last, halved

When numbers go up in equal steps, the average sits exactly in the middle: halfway between the first and the last.

Definition

For an evenly spaced list (the same gap every time), average = $\dfrac{\text{first} + \text{last}}{2}$ .
With an odd count that is the middle number; with an even count it is halfway between the two middle numbers.
Multiples of $k$ from $k$ up to $nk$ average $\dfrac{k(n + 1)}{2}$ .
For 'the last 6 multiples of 3 up to 60', write the first and last of that stretch (45 and 60), then halve their sum: 52.5.

Average of an evenly spaced list

\bar x = \dfrac{\text{first term} + \text{last term}}{2}

Worked example

Find the average of the first 9 multiples of 7.

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The same idea in a practice question:

Practice questionModerate

Example 4 · Average · Basic and Simple Averages

What is the average of last

12

even numbers from

2

40

Check the spacing first

1, 2, 4, 8 is not evenly spaced. Its average is

15 \div 4 = 3.75

, not

(1 + 8) \div 2 = 4.5

Find the right first term

The last 4 multiples of 6 up to 60 are 42, 48, 54 and 60. The stretch starts at 42, not at 6.

Drill 10 more on evenly spaced numbers: first plus last, halved

Concept 5 of 5: The first n natural, odd and even numbers

The natural, odd and even numbers are evenly spaced, so their averages come straight from the middle. Squares and cubes are not evenly spaced and need their sum formulas. Learn the rows below and most first-n questions take one line.

Definition

Run a formula backwards to find $n$ : if the first $n$ natural numbers average 12, then $\tfrac{n + 1}{2} = 12$ and $n = 23$ . Natural numbers start at 1. 'The first $n$ even numbers' means 2, 4, ..., $2n$ .

The first n ...	Sum	Average
Natural numbers 1 to n	$\dfrac{n(n + 1)}{2}$	$\dfrac{n + 1}{2}$
Odd numbers 1 to 2n − 1	$n^2$	$n$
Even numbers 2 to 2n	$n(n + 1)$	$n + 1$
Squares 1² to n²	$\dfrac{n(n + 1)(2n + 1)}{6}$	$\dfrac{(n + 1)(2n + 1)}{6}$
Cubes 1³ to n³	$\left(\dfrac{n(n + 1)}{2}\right)^2$	$\dfrac{n(n + 1)^2}{4}$

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The same idea in a practice question:

Practice questionModerate

Example 5 · Average · Basic and Simple Averages

The average of first

n

natural numbers is

15.5

. Find

n

Even numbers average n + 1, not n

The first 8 even numbers, 2 to 16, average 9. It is the odd numbers that average exactly

n

Squares are not evenly spaced

1, 4 and 9 average

\tfrac{14}{3}

, not

(1 + 9) \div 2 = 5

. Use the sum of squares.

Drill 27 more on the first n natural, odd and even numbers

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)

Average, total and count
Arithmetic mean
$\text{Average} = \dfrac{\text{Sum of the values}}{\text{Number of values}}$
A missing value, or the score needed for a target
The missing value
$x = A \times n - (\text{sum of the other } n - 1 \text{ values})$
When every value changes
A change applied to every value
$x_i \to a\,x_i + b \;\Rightarrow\; \bar x \to a\,\bar x + b$
Evenly spaced numbers: first plus last, halved
Average of an evenly spaced list
$\bar x = \dfrac{\text{first term} + \text{last term}}{2}$

Reference tables (1)

The first n natural, odd and even numbers5 rows

The first n ...	Sum	Average
Natural numbers 1 to n	$\dfrac{n(n + 1)}{2}$	$\dfrac{n + 1}{2}$
Odd numbers 1 to 2n − 1	$n^2$	$n$
Even numbers 2 to 2n	$n(n + 1)$	$n + 1$
Squares 1² to n²	$\dfrac{n(n + 1)(2n + 1)}{6}$	$\dfrac{(n + 1)(2n + 1)}{6}$
Cubes 1³ to n³	$\left(\dfrac{n(n + 1)}{2}\right)^2$	$\dfrac{n(n + 1)^2}{4}$

Watch out for (9)

An average of averages
→ Average, total and count
Zero is a value
→ Average, total and count
Count the test still to come
→ A missing value, or the score needed for a target
Order matters
→ When every value changes
Some values, not all
→ When every value changes
Check the spacing first
→ Evenly spaced numbers: first plus last, halved
Find the right first term
→ Evenly spaced numbers: first plus last, halved
Even numbers average n + 1, not n
→ The first n natural, odd and even numbers
Squares are not evenly spaced
→ The first n natural, odd and even numbers

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