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Average

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high importance~1 Q in Tier 116 formulas⚡ 12 shortcuts4 subtopics
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Subtopic 2 of 4·← Average & the sum bridge·Weighted average & two-group problems →

Members joining or leaving

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When a new member joins (or one leaves) and the average changes, the key quantity is the total sum, not the average.

  • New member whose entry changes the average from AA to A′A' (group of n): value =A′+n(A′−A)= A' + n(A' - A).
  • Member leaving: removed value =A′+n(A−A′)= A' + n(A - A'), where A' is the new average of the remaining n.
  • Replacement of one member: new − old =n(A′−A)= n(A' - A), i.e. replaced value =old+n(change)= \text{old} + n(\text{change}).

Equivalent view: the new member must carry the extra n(A′−A)n(A' - A) for all the old members plus his own share A′A'.

📖 Notes🧩 5 question types⚡ Formulas🎯 14 practice Q

Detailed notes

The situation

A group has an average. Then something happens: a person joins, a person leaves, one person is swapped for another, or a wrong value is found and corrected. The new average is given (or asked). The whole topic runs on one habit: work with totals, not averages.

Someone joins

Old group: n members, average A. After one person joins, the average of the n + 1 members is B. Newcomer=(n+1)B−nA=B+n(B−A)\text{Newcomer} = (n+1)B - nA = B + n(B - A) Think of it in words: the newcomer must bring his own share B plus the rise (B−A)(B - A) for each of the n old members. Example: 24 students average 15 years; with the teacher the average is 16. Teacher =16+24×1=40= 16 + 24 \times 1 = 40 years.

Someone leaves

n members average A; one leaves and the remaining n−1n - 1 average B. Leaver=nA−(n−1)B\text{Leaver} = nA - (n-1)B Example: 11 numbers average 36 (sum 396). After one is removed, 10 numbers average 34 (sum 340). Removed number =56= 56. If the average falls when 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 sum changes, and it changes by n×(change in average)n \times (\text{change in average}). New person=Old person+n×(change in average)\text{New person} = \text{Old person} + n \times (\text{change in average}) Example: the average weight of 8 people rises by 2.5 kg when a new person replaces one weighing 65 kg → new person =65+8×2.5=85= 65 + 8 \times 2.5 = 85 kg. If the average falls, subtract instead. If two people are replaced, the pair's total changes by n×n \times (change), so their average changes by that ÷ 2.

Wrong value corrected

A value was copied wrongly (84 written as 48). The sum is off by the difference, so Correct average=Wrong average+correct−wrongn\text{Correct average} = \text{Wrong average} + \frac{\text{correct} - \text{wrong}}{n} 30 students, average 62, with 84 read as 48: correct average =62+3630=63.2= 62 + \frac{36}{30} = 63.2. With two errors, add both corrections (with signs) before dividing.

A group joins or leaves

k new members join n old ones and the average becomes B: Sum of newcomers=(n+k)B−nA,their average=(n+k)B−nAk\text{Sum of newcomers} = (n+k)B - nA, \quad \text{their average} = \frac{(n+k)B - nA}{k} 30 students average 12 years; 10 join and the average becomes 13 → newcomers' sum =520−360=160= 520 - 360 = 160 → average 16 years.

Checking your answer

Always verify with totals: new total ÷ new count must give the stated new average. Options are often built from forgetting to multiply the change by n, or multiplying by n+1n + 1 in a replacement.

Quick revision

  • Join: newcomer =B+n(B−A)= B + n(B - A).
  • Leave: leaver =nA−(n−1)B= nA - (n-1)B.
  • Replace: new == old ±n×\pm n \times change.
  • Wrong entry: average changes by (correct − wrong)/n.
  • A group: work out the group's total from totals, then divide by its size.

Types of questions asked

Every way this subtopic shows up in exams — how to recognise it, the formula or logic to use, and a solved example.

Type 1: One member joins — find the newcomer's valuevery common2 practice Q
How to spot it:

A teacher, manager or new player joins a group; the average age/salary/weight goes up or down by a given amount.

x=B+n(B−A)=(n+1)B−nAx = B + n(B - A) = (n+1)B - nA
  1. Old total =nA= nA; new total =(n+1)B= (n+1)B.
  2. Newcomer == new total −- old total.
  3. Shortcut: newcomer =B+n×= B + n \times (rise). If the average falls, the rise is negative.

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

Example: The average age of 24 students is 15 years. Including the teacher, it becomes 16 years. Teacher's age?

16+24×1=4016 + 24 \times 1 = 40 years. (Check: 25×16−24×15=400−360=4025 \times 16 - 24 \times 15 = 400 - 360 = 40.)

Type 2: One member leaves — find the removed value or the new averagecommon2 practice Q
How to spot it:

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

removed=nA−(n−1)B\text{removed} = nA - (n-1)B
  1. Old total =nA= nA.
  2. Remaining total =(n−1)B= (n-1)B (or old total − removed value, if the value is given).
  3. Subtract to get the missing piece; divide if the new average is asked.

Why: removing a value only changes the sum and the count by known amounts.

Example: The average of 11 numbers is 36. After removing one number, the average of the rest is 34. Find the removed number.

11×36−10×34=396−340=5611 \times 36 - 10 \times 34 = 396 - 340 = 56.

Type 3: Replacement — one (or two) members swappedvery common3 practice Q
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.

new=old±n×d\text{new} = \text{old} \pm n \times d
  1. The count n stays the same.
  2. Change in total =n×= n \times change in average.
  3. New member == old member ++ that change (subtract if the average fell).

Why: only one value changed, so the whole change in the total belongs to it.

Example: The average weight of 8 persons rises by 2.5 kg when a new person replaces one weighing 65 kg. Weight of the new person?

65+8×2.5=8565 + 8 \times 2.5 = 85 kg.

Type 4: Wrongly recorded value correctedcommon2 practice Q
How to spot it:

'Later it was found that 84 was misread as 48' — the correct average is asked.

correct avg=wrong avg+correct−wrongn\text{correct avg} = \text{wrong avg} + \frac{\text{correct} - \text{wrong}}{n}
  1. For each error, find (correct value − wrong value), keeping the sign.
  2. Add these corrections.
  3. Divide by n and add to the wrong average.

Why: an error changes the total by exactly (correct − wrong) and nothing else.

Example: The average of 30 marks was 62, but 84 was read as 48. The correct average is?

62+84−4830=62+1.2=63.262 + \frac{84 - 48}{30} = 62 + 1.2 = 63.2.

Type 5: A group joins or leaves togethercommon2 practice Q
How to spot it:

k new students join (or k leave / k are swapped) and the new average is given; the group's average is asked.

group average=(n+k)B−nAk\text{group average} = \frac{(n+k)B - nA}{k}
  1. Old total =nA= nA; new total =(n+k)B= (n+k)B.
  2. The difference is the joining group's total.
  3. Divide by k for their average. For a swap of k members, total change =n×= n \times change.

Why: same total-bridge as a single member, just for k people at once.

Example: 30 students average 12 years. 10 new students join and the average becomes 13 years. Average age of the new students?

40×13−30×12=16040 \times 13 - 30 \times 12 = 160 → 16010=16\frac{160}{10} = 16 years.

Formulas

Joining member
value=A′+n(A′−A)\text{value} = A' + n(A' - A)
Leaving member
value=A′+n(A−A′)\text{value} = A' + n(A - A')
Replacement
new=old+n(A′−A)\text{new} = \text{old} + n(A' - A)
Count change both ways
total of newcomers=new total−old total\text{total of newcomers} = \text{new total} - \text{old total}

Shortcut tricks

⚡ Joining: own share + extras

New value = new average + (number of old members) × (average jump).

Example: The average age of 30 students is 14 years. When their teacher joins, the average becomes 15. Find the teacher's age.

Teacher = 15 + 30 × 1 = 45 years. (Check: 420 + 45 = 465 = 31 × 15 ✓)

⚡ Replacement: n × jump added

Only the swapped item changes the sum, by exactly n × (average change).

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

Jump = (44 − 20)/8 = 3 ⇒ new average = 24 + 3 = 27.

⚡ Exclusion: subtract the sums

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

Example: The average of 9 numbers is 27. If one number is excluded the average becomes 25. Find the excluded number.

9 × 27 − 8 × 25 = 243 − 200 = 43.

Where students lose marks

  • Using the OLD average where the NEW one is needed in the member formula.

  • Counting the new member inside n (n refers to the members BEFORE the change in the joining formula).

  • For replacement, computing the new number as old + change (missing the factor n).

  • Sign errors when the average falls instead of rising.

Practice sets — 14 questions

Sets of 10, mixed across the question types above. Each answer comes with a step-by-step explanation.

Topic test · 10 questions

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