Average

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high importance~1 Q in Tier 117 formulas⚡ 12 shortcuts4 subtopics

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Members joining or leaving

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⏱ 4 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

When a member joins or leaves and the average changes, work with totals, not averages.

Join: newcomer = new average + n × rise, with n the OLD count.
Leave: leaver = old total − new total of the rest.
Replace: new value = old value + n × (change in average). The count does not change.

A wrongly copied value shifts the total by (correct − wrong), so the average shifts by that ÷ n.

The habit: work with totals

A group has an average. Then one member joins, leaves, or is swapped out, and the average changes. Averages cannot be added or removed. Totals can.

Old total = old count × old average.
New total = new count × new average.
The joining or leaving value is the difference of the two totals.

Rule: Never move an average. Move the total (count × average), then divide at the end.

Someone joins

n members average A. One person joins, and the n + 1 members average B.

\text{newcomer} = (n+1)B - nA = B + n(B - A)

In words: the newcomer carries his own share B, plus the rise $B - A$ for each of the n old members.

Example: 24 students average 15 years. With the teacher included, the average becomes 16. Teacher = $16 + 24 \times 1 = 40$ years. Check: $25 \times 16 - 24 \times 15 = 40$ .

Someone leaves

n members average A. One leaves, and the remaining $n-1$ average B.

\text{leaver} = nA - (n-1)B

11 numbers average 36, so the sum is 396. One number is removed; the other 10 average 34, sum 340. Removed number = $396 - 340 = 56$ .

Watch: If the average falls after someone leaves, the leaver was above the old average. If it rises, the leaver was below it.

Someone is replaced

The count stays n. Only the total changes, and it changes by $n \times$ (change in average).

\text{new value} = \text{old value} + n \times (\text{change in average})

The average weight of 8 people rises by 2.5 kg when one person is replaced. The old person weighed 65 kg. New person = $65 + 8 \times 2.5 = 85$ kg. If the average falls, subtract instead. If TWO members are replaced, their pair's total changes by $n \times$ change — split it between them as the question directs.

A wrong entry is found

A value was copied wrongly — say 84 was read as 48. The total is off by exactly (correct − wrong).

\text{correct average} = \text{wrong average} + \frac{\text{correct} - \text{wrong}}{n}

30 students average 62 marks, with 84 read as 48. Correct average = $62 + \dfrac{36}{30} = 63.2$ . With two errors, add both corrections, with their signs, before dividing.

A whole group joins

k new members join n old ones, and the average becomes B.

\text{group total} = (n+k)B - nA, \qquad \text{group average} = \frac{(n+k)B - nA}{k}

30 students average 12 years. Ten new students join and the average becomes 13. Newcomers' total = $40 \times 13 - 30 \times 12 = 160$ . Their average = $160 \div 10 = 16$ years. If k members LEAVE instead, their total = $nA - (n-k)B$ — the same bridge, walked the other way.

Tip: Check every answer: new total ÷ new count must equal the stated new average.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common2 practice Q

One member joins — find the newcomer

How to spot it:

A teacher, manager or new player joins a group and the average age, salary or weight moves by a given amount.

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

Method

Write the old total nA and the new total (n+1)B.
Subtract: the newcomer is the difference.
Shortcut: newcomer = B + n × rise. If the average falls, the rise is negative.

Why it works:

The newcomer supplies his own share and lifts every old member by the change.

Try this

The average age of 24 students is 15 years. Including their teacher, it becomes 16 years. Find the teacher's age.

Show solution

Newcomer = $16 + 24 \times 1$ .
= $40$ years.
Check: $25 \times 16 - 24 \times 15 = 400 - 360 = 40$ .

Answer

40 years

Type 2common2 practice Q

One member leaves — the removed value

How to spot it:

'One number is excluded / one member leaves, and the average of the rest becomes …'.

\text{removed} = nA - (n-1)B

Method

Old total = nA.
Remaining total = (n − 1)B.
Subtract the two totals; divide later only if the new average is asked instead.

Why it works:

Removing one value changes the sum and the count by known amounts, and nothing else.

Try this

The average of 12 numbers is 40. One number is removed and the average of the remaining numbers is 38. Find the removed number.

Show solution

Old total = $12 \times 40 = 480$ .
New total = $11 \times 38 = 418$ .
Removed = $480 - 418 = 62$ .

Answer

Type 3very common3 practice Q

One member is replaced

How to spot it:

'A new man replaces one weighing 65 kg and the average increases by 2.5 kg.' The count does not change.

\text{new} = \text{old} \pm n \times d

Method

Keep the count n the same.
Change in total = n × change in average.
New member = old member + that change (subtract if the average fell).

Why it works:

Only one value changed, so the whole change in the total belongs to it.

Try this

The average weight of 9 persons falls by 2 kg when a person weighing 78 kg is replaced by a newcomer. Find the newcomer's weight.

Show solution

Total falls by $9 \times 2 = 18$ kg.
Newcomer = $78 - 18 = 60$ kg.

Answer

60 kg

Type 4common2 practice Q

A wrongly recorded value is corrected

How to spot it:

'Later it was found that 84 had been misread as 48.' The correct average is asked.

\text{correct avg} = \text{wrong avg} + \frac{\text{correct} - \text{wrong}}{n}

Method

For each error, work out (correct value − wrong value), keeping the sign.
Add the corrections.
Divide by n and add to the wrong average.

Why it works:

An error changes the total by exactly (correct − wrong) and nothing else.

Try this

The average of 25 marks was calculated as 48. Later, 72 was found to have been misread as 27. Find the correct average.

Show solution

Correction = $72 - 27 = 45$ .
Shift = $45 \div 25 = 1.8$ .
Correct average = $48 + 1.8 = 49.8$ .

Answer

49.8

Type 5common2 practice Q

A whole group joins or leaves

How to spot it:

k new students join (or k leave), the new overall average is given, and the group's own average is asked.

\text{group average} = \frac{(n+k)B - nA}{k}

Method

Old total = nA; new total = (n + k)B.
The difference is the joining group's total.
Divide by k for their average. For a swap of k members, the total changes by n × change.

Why it works:

The same total bridge as for a single member, taken k people at once.

Try this

30 students have an average age of 12 years. 10 new students join and the average becomes 13 years. Find the average age of the new students.

Show solution

New total = $40 \times 13 = 520$ ; old total = $30 \times 12 = 360$ .
Newcomers' total = $520 - 360 = 160$ .
Their average = $160 \div 10 = 16$ years.

Answer

16 years

Formula sheet

Joining member

\text{value} = A' + n(A' - A)

n = old count; A' = new average.

Leaving member

\text{value} = A' + n(A - A')

A' = average of the remaining n members.

Replacement

\text{new} = \text{old} + n(A' - A)

The count does not change.

Count change both ways

\text{total of newcomers} = \text{new total} - \text{old total}

Shortcuts that save time

⚡ Joining: own share plus extras

New value = new average + (old count) × (average jump).

Example

The average age of 30 students is 14 years. Their teacher joins them and the average becomes 15 years. Find the teacher's age.

Show solution

Jump in average = $15 - 14 = 1$ year.
Teacher = $15 + 30 \times 1 = 45$ years.
Check: $31 \times 15 - 30 \times 14 = 465 - 420 = 45$ .

Answer

45 years

⚡ Replacement: the jump times n

Only the swapped item changes the sum, and it changes by n × (average jump).

Example

The average of 8 numbers is 24. If 20 is replaced by 44, find the new average.

Show solution

Sum rises by $44 - 20 = 24$ .
Jump = $24 \div 8 = 3$ .
New average = $24 + 3 = 27$ .

Answer

⚡ Exclusion: subtract the totals

Old total − (new count × new average) = the removed value.

Example

The average of 9 numbers is 27. One number is excluded and the average of the rest becomes 25. Find the excluded number.

Show solution

Old total = $9 \times 27 = 243$ .
New total = $8 \times 25 = 200$ .
Excluded = $243 - 200 = 43$ .

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using the old average where the new one belongs.

Label both averages first: A old, B new. The formula reads off B for shares.

Mistake 02

Counting the newcomer inside n.

In the joining formula, n is the count BEFORE the newcomer arrives.

Mistake 03

For a replacement, changing the new number by just the average jump.

The swap changes the total by n × jump, so the new value moves by n × jump.

Mistake 04

Subtracting when the average rises (or adding when it falls).

Fix the sign from the story: rise = plus, fall = minus, for joiners and replacements alike.

Mistake 05

Correcting a wrong entry by editing the average directly.

Adjust the total by (correct − wrong), then divide by n.

Quick revision

Read this the night before the exam.

Join: newcomer $= B + n(B - A)$ , n = old count.
Leave: leaver $= nA - (n-1)B$ .
Replace: new = old ± n × (change in average).
Wrong entry: average shifts by (correct − wrong) ÷ n.
Group joins: group total = (n+k)B − nA, then ÷ k for their average.
Check: new total ÷ new count = stated new average.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 6 min · wrong answers go to your mistake notebook automatically.

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