📑 On this page
⚡ The short answer
When someone joins or leaves, work with totals, not averages: total = average × count. Add the joiner (or subtract the leaver), then divide by the new count. Shortcut: the mover's value = new average + old count × the change. The trap: forgetting that the count changes too.
🎬 The 30-second story
The screen-time squad
1
“Squad average: 6 hours a day. Embarrassing.”
📱
Zoya
Five friends, one tracker: 30 hours a day.
2
“I'm out. Digital detox, starting now.”
🧘
Aarav
Aarav, the 11-hour doomscroller, quits.
3
“We dropped to 4¾ hours and nobody changed!”
📉
Meher
19 hours, now shared by 4.
4
“Add me. I do exactly 4¾ hours.”
➕
Rohan
Rohan joins. Nothing moves.
Tara's maths: An average moves by how far the mover sits from it, shared across the group. Aarav was 5 above 6; removing him costs the 4 who stay 1.25 each. Rohan sits on the average, so nothing moves.
What happens to the average when someone joins or leaves?
The whole topic in one line: an average is a total in disguise. Six hours across five friends is really 30 hours, split equally. CLAT never gives individual values, so fix the total, fix the count, divide again.
🧮 The formula
The only recipe you need
🧮
Old total
old average × old count
±
🚪
The mover
add a joiner, subtract a leaver
=
⚖️
New total
÷ new count = new average
⚔️ Side by side
Joining vs leaving at a glance
🙋Someone joins
count goes up by 1
- •Above the average: lifts it
🚶Someone leaves
count goes down by 1
- •Above the average: drags it down
VS
| Test | Someone joins | Someone leaves |
|---|---|---|
| New total | old + joiner | old − leaver |
| New count | n + 1 | n − 1 |
| Mover = average | no change | no change |
Average fell when someone left? They were above it.
What is the formula for the new average?
With old count n, old average A and mover x: the new average is (n × A + x) ÷ (n + 1) when x joins and (n × A − x) ÷ (n − 1) when x leaves. That is the recipe in symbols.
📌 The surplus idea: why the shortcut works
The mover's surplus is how far x sits from the old average. Everyone else already sits at A, so only the surplus is new, shared by the new count.
Flip it: mover = new average + old count × change (rise for a joiner, fall for a leaver). Its twin, old average + new count × change, agrees.
📊 The data
Same newcomer, different group sizes
Joins a group of 4
4 marks
Joins a group of 9
2 marks
Joins a group of 19
1 marks
Joins a group of 99
0.2 marks
Someone 20 above the average joins. Lift = 20 ÷ new count.
How do you find the value of the person who joined or left?
CLAT's favourite reverse question: the average changed, so what did the mover bring?
✍️ Worked example
Warm-up
⏱ 30 sec
Meher joins the study group
A study group of 12 averages 74 in a mock. Meher, who scored 87, joins. New average?
- 1Old total = 12 × 74 = 888.
- 2Add Meher: 888 + 87 = 975.
- 3New count = 12 + 1 = 13.
- 4New average = 975 ÷ 13 = 75.
Answer: New average = 75
⚡ The exam-hall shortcut
Meher is 13 above the average. 13 marks shared by 13 people: +1. So 75.
✍️ Worked example
CLAT level
⏱ 40 sec
Who walked out of the gym batch?
A gym batch of 15 averages 72 kg. One quits; the rest average 71 kg. The quitter's weight?
- 1Old total = 15 × 72 = 1,080 kg.
- 2New total = 14 × 71 = 994 kg.
- 3Leaver = 1,080 − 994 = 86 kg, above 72 as the fall predicted.
Answer: 86 kg
⚡ The exam-hall shortcut
71 + 15 × 1 = 86, or 72 + 14 × 1 = 86. Two doors, one room.
✍️ Worked example
CLAT level
⏱ 30 sec
The collab swap
Eight creators average 4,500 followers. One with 1,900 is swapped out and the average rises by 250. The newcomer's followers?
- 1The count stays 8, so this is a replacement.
- 2Rise in total = 8 × 250 = 2,000, all from the swap.
- 3Newcomer = 1,900 + 2,000 = 3,900.
Answer: 3,900 followers
⚡ ⏱ Replacement rule
Incoming − outgoing = count × change. The 4,500 was never used: CLAT loves a decoy number.
Is there a 10-second shortcut for average change questions?
Yes: ask which way the average moved and what the new count is. Direction alone often kills two options.
✍️ Worked example
Tricky
⏱ 60 sec
Two hypotheticals, one group
If Aarav (90) joins a quiz team, its average rises by 2. If instead Zoya (30) joins, it falls by 2. Find the team's size and average.
- 1Joiner = old average + new count × change: 90 = A + 2(n + 1) and 30 = A − 2(n + 1).
- 2Add: 120 = 2A, so A = 60. Subtract: 60 = 4(n + 1), so n = 14.
- 3Check: 930 ÷ 15 = 62 and 870 ÷ 15 = 58.
Answer: 14 people, average 60.
⚡ The exam-hall shortcut
Equal and opposite shifts put the average midway between 90 and 30: 60. A 30-point surplus moves it by 2, so the new count is 15.
🎛️ Try it yourself — change any number
Try it: the judo team
Current total
680
Average if it joins
69.23
Average if it leaves
66.5
Total = 10 × 68 = 680; then add or remove 81.5 and divide by the new count
CLAT 2023's judo numbers: the 'leaves' row shows 66.5, as in the passage. Type 68 as the mover: nothing moves.
💡 Estimate before you calculate
Average fell when someone left? The leaver was above it: strike every option at or below. In the judo set, that kills two.
How does CLAT hide this inside a caselet?
CLAT 2023 wrapped this topic in a judo selection: ten weight groups, one player from each, an average of 68 kg, and a player S who walks out. Three of its five questions are pure join-and-leave maths.
| Group | Range (kg) | Group | Range (kg) |
|---|---|---|---|
| 1 | 48–52 | 6 | 68–72 |
| 2 | 52–56 | 7 | 72–76 |
| 3 | 56–60 | 8 | 76–80 |
| 4 | 60–64 | 9 | 80–84 |
| 5 | 64–68 | 10 | 84–88 |
→ scroll to see more
- 1
Averages into totals10 × 68 = 680; 9 × 66.5 = 598.5.
- 2
Find the moverS = 680 − 598.5 = 81.5 kg: group 9.
- 3
Build the next total11 × 68 = 748, so the two joiners weigh 149.5 kg.
- 4
Test the rangesAdd each pair's lowest and highest weights. Does 149.5 fit?
📊 The data
The judo caselet as four totals
10 players (all groups)
680 kg
9 players (S has left)
598.5 kg
11 players (2 joined)
748 kg
The two newcomers
149.5 kg
Each question is a subtraction between two bars: S = 680 − 598.5 = 81.5 kg.
⚠️ Ranges, not single numbers
Joiners from weight bands give intervals: players from 60–64 and 84–88 weigh 144 to 152 kg together. The bands also share end-points (52 kg sits in groups 1 and 2); CLAT's answers avoided the edges, but flag it if yours lands on one.
What traps does CLAT set on joining and leaving?
- ✓The forgotten count. Dividing the new total by the old count.
- ✓Surplus over the wrong count. A joiner's surplus is shared by the new group, joiner included.
- ✓Swap read as a join. In a swap the count is unchanged.
- ✓The missing count. One condition, two unknowns: often 'cannot be determined'.
- ✓The decoy number. Data the question never needs.
🧭 Decision flow
Which rule do I use?
❓ Did the number of members change?
No →
Swap: incoming − outgoing = count × change
Yes ↓
❓ Do you know the old count?
No →
Probably 'cannot be determined'
Unless two scenarios are given (2018 Q141).
Yes ↓
❓ Is it exactly one person moving?
Yes →
Mover = new average + old count × change
Rise for a joiner, fall for a leaver.
No →
Several movers: new total − old total
How did CLAT ask about adding and removing values?
📝 Asked in CLAT 2023
· Q146
Who left?
Players are selected for Judo based on their body weights from the following 10 weight groups: 1. (48 kg - 52 kg) 2. (52 kg - 56 kg) 3. (56 kg - 60 kg) 4. (60 kg - 64 kg) 5. (64 kg - 68 kg) 6. (68 kg - 72 kg) 7. (72 kg - 76 kg) 8. (76 kg - 80 kg) 9. (80 kg - 84 kg) 10. (84 kg - 88 kg). The average weight of the players after selecting one player from each group is 68 kg. If one of the players (named S) leaves the team, their average weight comes down to 66.5 kg.
Player S is from the weight group:
A1
B9
C5
D10
Reveal answer + reasoning
Answer: B
10 × 68 = 680; 9 × 66.5 = 598.5. S = 81.5 kg, in group 9 (80–84).Fast: 66.5 + 10 × 1.5 = 81.5. And since the average fell, S was above 68, so A and C die on sight. Time: 30 seconds.
Why the others fail
- A — Removing a 48–52 kg player would raise the average.
- C — Group 5 tops out at 68, the average; such a leaver can't lower it.
- D — 84 kg or more would drop the average by over 1.5 kg.
📝 Asked in CLAT 2023
· Q147
One out, two in
Players are selected for Judo based on their body weights from the following 10 weight groups: 1. (48 kg - 52 kg) … 3. (56 kg - 60 kg) 4. (60 kg - 64 kg) 5. (64 kg - 68 kg) … 7. (72 kg - 76 kg) … 9. (80 kg - 84 kg) 10. (84 kg - 88 kg) The average weight of the players after selecting one player from each group is 68 kg. If one of the players (named S) leaves the team, their average weight comes down to 66.5 kg.
If S leaves the group and two new players join the group, their average weight increases to 68 kg. These players can NOT be from groups:
A1 and 3
BBoth from group 7
C4 and 10
D5 and 9
Reveal answer + reasoning
Answer: A
9 players weigh 598.5; 11 at 68 weigh 748; so the pair weighs 149.5 kg.1 and 3 span only 104 to 112: impossible. The other three pairs each span 144 to 152. The stem says NOT. Time: 60 seconds.
Why the others fail
- B — 74.75 + 74.75 = 149.5. Possible.
- C — 62 + 87.5 = 149.5. Possible; a pick for those who miss NOT.
- D — 66 + 83.5 = 149.5. Possible.
📝 Asked in CLAT 2023
· Q150
Find the partner
Players are selected for Judo based on their body weights from the following 10 weight groups: … 4. (60 kg - 64 kg) 5. (64 kg - 68 kg) … 7. (72 kg - 76 kg) … 10. (84 kg - 88 kg) The average weight of the players after selecting one player from each group is 68 kg. If one of the players (named S) leaves the team, their average weight comes down to 66.5 kg. [Q147: S leaves, two new players join, and the average becomes 68 kg.]
If one of the new two players is from group 4, which group the other player is from?
A5
B7
C10
DNone of the above
Reveal answer + reasoning
Answer: C
The pair weighs 149.5 kg (Q147). Group 4 is 60–64, so the other weighs 85.5 to 89.5 kg. Only group 10 (84–88) overlaps. Time: 30 seconds.
Why the others fail
- A — 64 + 68 = 132 at most.
- B — 64 + 76 = 140 at most.
- D — Group 10 works: 62 + 87.5 = 149.5.
📝 Asked in CLAT 2018
· Q141
Two scenarios, same average
The number of children in a camp is x and their average weight is 20 kg. If 5 children each weighing 12 kg, join the camp or if 10 children each weighing 21 kg leave the camp, the average weight in both the cases remain the same.
The value of x is
A18
B16
C15
D14
Reveal answer + reasoning
Answer: C
5 joiners each 8 below 20: a 40 kg deficit over x + 5. 10 leavers each 1 above 20: a 10 kg loss over x − 10. Equal drops: 40 ÷ (x + 5) = 10 ÷ (x − 10), so 30x = 450 and x = 15.Check: 360 ÷ 20 = 18 = 90 ÷ 5. Time: 45 seconds, or plug options.
Why the others fail
- A — 18.26 vs 18.75: unequal.
- B — 18.10 vs 18.33: unequal.
- D — 17.89 vs 17.5: unequal.
What do students get wrong about average shifts?
💥 Myth vs fact
Traps students fall for
🙅 Myth
A topper joining lifts the average a lot.
Tap to bust it →
✅ Fact
Lift = surplus ÷ new count. Joining 99 people, a 50-mark surplus adds 0.5.
🙅 Myth
The average fell 1.5, so the leaver was 1.5 above it.
Tap to bust it →
✅ Fact
He was old count × fall above the new average: 66.5 + 15 = 81.5.
🙅 Myth
Same average means same group.
Tap to bust it →
✅ Fact
Judo Q147: one left, two joined, still 68.
🙅 Myth
You need everyone's value to find the new average.
Tap to bust it →
✅ Fact
Total and count carry everything.
Test yourself: five average-shift questions
🎯 Quick quiz
Join, leave, swap
0 / 5
20 songs average 3.5 minutes. A 5.6-minute track is added.
Q1. What is the new average length?
Surplus 2.1 ÷ 21 = 0.1, so 3.6.25 hostel students average ₹1,800 a month on food delivery. One moves out; the rest average ₹1,750.
Q2. What did the leaver spend?
1,750 + 25 × 50 = 3,000.A 6-member debate team averages 70. A member who scored 58 is replaced; the average rises by 3.
Q3. What did the new member score?
58 + 6 × 3 = 76.A 40-year-old coach joins some trainees; the average age rises by 1 year.
Q4. What was the trainees' average age?
A = 39 − n. Unknown n: 30 if n = 9, 35 if n = 4.A class of 40 averages 55. Two latecomers take the test; the average becomes 56.
Q5. What do the latecomers average?
2,352 − 2,200 = 152, so 76 each on average.🎯 Adding and removing values in a nutshell
- An average is a total in disguise: total = average × count.
- Join: count + 1. Leave: count − 1. Swap: count unchanged.
- Change in average = surplus ÷ new count.
- Mover = new average + old count × change.
- No count, one condition: 'cannot be determined'.
📎 Formula card
- Total = average × count.
- Join x: (n × A + x) ÷ (n + 1). Leave x: (n × A − x) ÷ (n − 1).
- Joiner = new average + old count × rise.
- Leaver = new average + old count × fall.
- Swap: incoming − outgoing = count × change.
- k join averaging y: (n × A + k × y) ÷ (n + k).
Frequently asked questions
How do you find the weight of a person who left a group?
Old average × old count gives the old total; new average × new count gives the new total. The difference is the leaver's weight. Faster: new average plus old count times the fall. If 10 players average 68 kg and 66.5 kg after one leaves, he weighed 66.5 + 10 × 1.5 = 81.5 kg.
What happens to the average when a new person joins?
It moves towards the newcomer: up if they are above the average, down if below, unchanged if equal. The change is the newcomer's distance from the old average divided by the new count, so someone 13 marks above the average of 12 people lifts it by exactly 1.
How do you find the new value when one member is replaced?
In a swap the count stays the same, so the whole change in total comes from the swap: incoming minus outgoing equals count times the change in average. If a team of 6 averages 3 more after a 58-scorer is replaced, the newcomer scored 58 + 6 × 3 = 76.
Can a CLAT average-change question be 'cannot be determined'?
Yes. If you only know the average rose when one person joined, and have neither the count nor a second condition, many answers fit. Two unknowns, the count and the average, need two facts. CLAT 2018 Q141 gave two scenarios precisely so the count could be found.
Keep going
← Previous topic
Average: Formula, Meaning and the Balance Idea
Average = fair share: the formula, the balance idea and CLAT traps
Next topic →
Weighted Average: Why You Can't Just Average the Averages
Why averaging two averages fails, and the see-saw fix CLAT rewards
📚
Part of the chapter
Averages, Mixtures & Alligation for CLAT Quantitative Techniques
See how this topic fits the whole of Averages, Mixtures & Alligation →
Connected topics in other chapters
Average: Formula, Meaning and the Balance Idea
The sum = average × count idea this page is built on.
Weighted Average
When a whole group joins, not one person, you are combining averages.
Mixtures, Concentration and Replacement
Removing and adding litres of a solution is the same totals logic.
Caselet DI: Turning a CLAT Quant Passage into a Table
How to turn a passage like the judo set into totals you can subtract.
Problems on Ages
Average-age questions where a member joins or leaves a family.
Every topic in Averages, Mixtures & Alligation
- 1.Average: Formula, Meaning and the Balance Idea
- 2.When Someone Joins or Leaves: How the Average Shifts (you're here)
- 3.Weighted Average: Why You Can't Just Average the Averages
- 4.Alligation: the Cross Method
- 5.Mixtures, Concentration and Replacement
- 6.Average Speed: Why It Is Not the Average of Speeds