AFCAT Numerical Ability · Average

Groups: Combine Totals, Split Totals, Watch the Overlap

Turn every group's average into a total. Totals add to give a combined average, subtract to give the average of the rest, and a value two groups share is counted twice.

All Average notes

Why this matters

Four AFCAT past-year questions are on this page, from 2018 and 2019, and about a fifth of it is HARD. Half of the HARD ones are overlaps: when 'the first few' and 'the last few' of a list share a number, that number sits in both totals.

Concept 1 of 4: Combining groups

The average of a whole class is not the plain average of its two sections' averages. The bigger section counts for more, so add the totals and divide by everyone.

Definition

Combined average = sum of the group totals ÷ total count.
It always lies between the group averages, nearer the bigger group.
Balance view: the amount above the combined average equals the amount below it. If 6 people earn ₹9,000 when the overall average is ₹7,000, they are $6 \times 2000 = 12{,}000$ above. If everyone else earns ₹6,000 (1,000 below), there must be $12{,}000 \div 1000 = 12$ others.

Combined average of two groups

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

Worked example

Section A has 20 students averaging 64 and section B has 16 averaging 73. Find the class average.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Average · Combined and Group Averages

The average of 30 numbers is 40 and that of another 40 numbers is 30. The average of all the numbers is

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.

Drill 10 more on combining groups

Concept 2 of 4: The rest of the group

Subtract what you know from the whole total. What is left belongs to the part you do not know.

Definition

Total of the rest = whole total − the totals of the known parts. Divide by how many are left.
With three parts, subtract both known totals.
A single unknown value is a 'group' of one: the 8th of 12 numbers, between the first 7 and the last 4, is the whole total minus those two totals.
Money stories: income = expenditure + savings, in total or month by month.

Average of the rest

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

Worked example

The average of 14 numbers is 24. The first 4 average 21 and the next 8 average 26. Find the average of the last 2.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

AFCAT · 2019 · AFCAT (I) 2019 — Numerical Ability · 02-203Easy

Example 2 · Average · Combined and Group Averages

The average of 9 numbers is 40. The average of the first 5 numbers is 20 and that of the last 3 numbers is 35. What is the 6th number?

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.

Drill 35 more on the rest of the group

Concept 3 of 4: Overlapping groups: the shared value counted twice

'The first 6 and the last 6' of 11 numbers cover 12 places, so one number, the 6th, is in both groups. Adding the two totals counts it twice; taking away the whole total leaves exactly that number.

Definition

Shared count = $n_1 + n_2 - n$ . If it is 1, the formula gives that value; if it is 2 or more, it gives their sum.
If $n_1 + n_2 < n$ , the groups do not meet. There is a gap instead, and the 'rest of the group' method applies.

The shared value

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

Worked example

The average of 11 numbers is 50. The first 6 average 49 and the last 6 average 52. Find the 6th number.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Average · Combined and Group Averages

The average of 13 numbers is 40. If the average of the first 7 numbers is 42 and that of the last 7 numbers is 35, find the 7th number.

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.

Drill 11 more on overlapping groups: the shared value counted twice

Concept 4 of 4: Pair averages: add all three

Each person appears in two of the three pairs. Add the three pair totals and you have counted everyone twice.

Definition

Turn each pair average into a pair total (× 2).
Half the sum of the three pair totals is $a + b + c$ .
Each value = that full total − the pair it is missing from.
Overlapping runs of days work the same way: Monday to Wednesday and Tuesday to Thursday share two days, so Thursday − Monday is the difference of the two totals.

Three values from three pairs

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

Worked example

The average of P and Q is 30, of Q and R is 34, and of P and R is 38. Find Q.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Average · Combined and Group Averages

The average of

A

and

B

is 20, the average of

B

and

C

is 19 and the average of

C

and

A

is 21. Find the value of

A

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.

Drill 2 more on pair averages: add all three

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)

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

Watch out for (4)

Not the plain mean of the averages
→ Combining groups
Check that the groups cover everyone once
→ The rest of the group
The overlap may be more than one number
→ Overlapping groups: the shared value counted twice
Halve the sum
→ Pair averages: add all three

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