Average — AFCAT Numerical Ability Formulas and Traps

The Average, Its Total and the First n Numbers

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}

The first n natural, odd and even numbers

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	$\left(\dfrac{n(n + 1)}{2}\right)^2$	$\dfrac{n(n + 1)^2}{4}$

Common traps

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.

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.

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.

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.

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.

Consecutive Numbers: the Middle Term Does the Work

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An odd count: the average is the middle number

Ends of an odd-length run

\text{largest} = A + \dfrac{n - 1}{2}\,d, \qquad \text{smallest} = A - \dfrac{n - 1}{2}\,d

An even count: halfway between the two middle numbers

Ends of an even-length run

\text{largest} = A + \dfrac{n - 1}{2}\,d \quad (n \text{ even, so } \tfrac{n - 1}{2} \text{ ends in } .5)

Pairs from the two ends

End pairs and spread

\text{first} + \text{last} = 2\bar x, \qquad \text{last} - \text{first} = (n - 1)\,d

Letters, products, and a run that changes

A run centred on its average

x - 2,\; x - 1,\; x,\; x + 1,\; x + 2 \quad (\text{five terms averaging } x)

Common traps

Use the right step

Five consecutive odd numbers averaging 31 run from 27 to 35, not from 29 to 33. Odd and even numbers step by 2.

Check the parity

Six consecutive odd numbers can never average 25: an even count of odd numbers averages an even number. A question like that cannot be solved as printed.

The spread does not depend on the average

Any 12 consecutive integers span 11 from the smallest to the largest, not 12: there are 11 gaps between 12 numbers.

Centre the run, do not start it

Writing five consecutive numbers as

x, x + 1, \dots, x + 4

makes

x

the smallest, not the average. Centring them on

x

keeps the average equal to

x

.

One Value In, Out or Changed

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One more value: a new day, a teacher, a manager

One value added

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

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}

Common traps

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.

A low value leaving raises the average

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

(550 - 30) \div 10 = 52

.

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

.

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

.

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.

Groups: Combine Totals, Split Totals, Watch the Overlap

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Combining groups

Combined average of two groups

\bar x = \dfrac{n_1A_1 + n_2A_2}{n_1 + n_2}

The rest of the group

Average of the rest

A_{\text{rest}} = \dfrac{nA - n_1A_1}{n - n_1}

Overlapping groups: the shared value counted twice

The shared value

\text{shared value(s)} = n_1A_1 + n_2A_2 - nA

Pair averages: add all three

Three values from three pairs

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

Common traps

Not the plain mean of the averages

A group of 10 averaging 50 and a group of 50 averaging 80 have an overall average of

(500 + 4000) \div 60 = 75

, not 65.

Check that the groups cover everyone once

If the first 7 of 12 numbers and the last 4 are known, one number (the 8th) is left over. Count the places before you subtract.

The overlap may be more than one number

The first 12 and the last 10 of 20 numbers share two numbers, so the formula gives their sum, not a single value.

Halve the sum

The three pair totals add up to twice the total. If they come to 204, then

P + Q + R = 102

, not 204.

Groups That Join or Leave

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A group joins

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

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

Common traps

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.

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.

Two changes to the count

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

32 - 9 = 23

or

32 + 7 = 39

gives a wrong average.

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

.

Weighted Averages, Rates and Percentages

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Weights given as numbers or ratios

Weighted average

\bar x = \dfrac{w_1x_1 + w_2x_2 + \dots}{w_1 + w_2 + \dots}

Averages of rates: speed, pass rate, return

Average speed

\text{average speed} = \dfrac{\text{total distance}}{\text{total time}}

Percentage shares as weights, and the missing share

Overall average from percentage shares

\bar x = \dfrac{p_1x_1 + p_2x_2 + \dots}{100}

Values a percentage above or below the average

A value relative to the average

\text{value} = A\left(1 \pm \dfrac{p}{100}\right)

Common traps

Divide by the weights, not the count

60 and 80 with weights 1 and 3 average

(60 + 240) \div 4 = 75

. Dividing 300 by 2 values gives 150, which is not even between them.

Speeds do not average directly

Going out at 30 km/h and back the same way at 50 km/h averages 37.5 km/h, not 40, because more time is spent at the slower speed.

Turn the percentage into people

'15% of 40 students failed' means 6 students. Use 6 as the count of that group, not 15.

Which average?

'20% above the group average' uses the average before anyone leaves or joins, unless the question says otherwise. Working it out from the new average gives a different number.

The Highest, the Lowest, and What Is Left

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Removing the highest and the lowest

The two extremes together

H + L = nA - (n - 2)A'

One extreme alone: the largest or the smallest

One value taken out

x = nA - (n - 1)A'

Highest and lowest from their sum and difference

Two numbers from sum and difference

H = \dfrac{(H + L) + (H - L)}{2}, \qquad L = \dfrac{(H + L) - (H - L)}{2}

Common traps

Count what is left

Removing the highest and the lowest of 20 leaves 18, so the new total is 18 times the new average, not 20 times.

Check that it really is the extreme

If 6 numbers average 30 and leaving out 'the smallest' gives an average of 26, the number removed is

180 - 130 = 50

. That cannot be the smallest, so re-read the question.

Halve at the end

With

H + L = 144

and

H - L = 70

, adding gives

2H = 214

, so

H = 107

. Stopping at 214 is the slip.

Batting Averages, Runs Needed and Run Rates

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Batting average and the next innings

Batting average

\text{batting average} = \dfrac{\text{runs}}{\text{innings}} \quad \left(\text{or } \dfrac{\text{runs}}{\text{dismissals}}\right)

Runs needed for a target, and how many innings

Score needed in the next innings

x = A'(n + 1) - An

Run rate, required rate and economy

Run rates

\text{run rate} = \dfrac{\text{runs}}{\text{overs}}, \qquad \text{required rate} = \dfrac{\text{target} - \text{runs so far}}{\text{overs left}}

Common traps

Not-outs change the divisor

1,000 runs in 25 innings with 5 not-outs is a batting average of

1000 \div 20 = 50

, not

1000 \div 25 = 40

.

Count the innings still to come

To average 33 after 17 innings he needs

17 \times 33

runs in all, not

16 \times 33

.

Do not average two rates

12 overs at 5 an over and 4 overs at 9 make 96 runs in 16 overs, a rate of 6, not

(5 + 9) \div 2 = 7

.

Ages Over Time, and a New Baby

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Moving a family through time

A family's total after t years

\text{total after } t \text{ years} = \text{total now} + n\,t

A child or a baby joins the family

A child joins

\text{new average} = \dfrac{\text{family total} + \text{child's age}}{n + 1}

Finding the baby's age

The baby's age

\text{baby} = (n + 1)A_{\text{now}} - \left(nA_{\text{then}} + n\,t\right)

Births at different times

Today's total

\text{total now} = \text{past total} + n\,t + (\text{each child's age now})

Common traps

The gap stays, the ratio does not

A mother 24 years older than her daughter is always 24 years older, but 'twice as old' is true for only one year.

Age the family first

If the baby arrives 3 years later, the five older members are 15 years older in total. Adding the baby to the old total misses those 15 years.

Age the older members first

If 6 members totalled 120 four years ago, they total 144 now. Forgetting those 24 years makes the baby 24 years too old.

A just-born baby is 0

If a question says a baby is born 'today' and the arithmetic gives the baby an age of 3, the question is inconsistent. A newborn adds nothing to the total.

Children age from their own birth

A child born 6 years ago adds 6 to today's total, not the 8 years since the old average was taken.

Average formulas

The Average, Its Total and the First n Numbers

Average, total and count

A missing value, or the score needed for a target

When every value changes

Evenly spaced numbers: first plus last, halved

The first n natural, odd and even numbers

Common traps

Consecutive Numbers: the Middle Term Does the Work

An odd count: the average is the middle number

An even count: halfway between the two middle numbers

Pairs from the two ends

Letters, products, and a run that changes

Common traps

One Value In, Out or Changed

One more value: a new day, a teacher, a manager

Finding the value that joined or left

Replacing a member

A value misread or miscopied

Common traps

Groups: Combine Totals, Split Totals, Watch the Overlap

Combining groups

The rest of the group

Overlapping groups: the shared value counted twice

Pair averages: add all three

Common traps

Groups That Join or Leave

Some leave, others join

Finding the group that joined or left

Common traps

Weighted Averages, Rates and Percentages

Weights given as numbers or ratios

Averages of rates: speed, pass rate, return

Percentage shares as weights, and the missing share

Values a percentage above or below the average

Common traps

The Highest, the Lowest, and What Is Left

Removing the highest and the lowest

One extreme alone: the largest or the smallest

Highest and lowest from their sum and difference

Common traps

Batting Averages, Runs Needed and Run Rates

Batting average and the next innings

Runs needed for a target, and how many innings

Run rate, required rate and economy

Common traps

Ages Over Time, and a New Baby

Moving a family through time

A child or a baby joins the family

Finding the baby's age

Births at different times

Common traps

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