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Average

Lesson 13 of 113

1. Ye topic hai kya?

Average (arithmetic mean) = total ko kitne pieces mein baanta. avg = (sum) / n. Paper teen ghumav: (1) list / pehle n natural, (2) ek innings add/remove / galat figure, (3) do groups milao ya equal-distance speed. Statistics wala class-interval, median class, ogive yahan nahi. Cricket “40 chahiye, 6th kitna” yahin hai — Statistics ne yahi bheja tha.

Paper yahan kya maangta

Sum = avg × n. Pehle n natural / odd / even. Naya number add. Innings se target. Combined (n₁a₁+n₂a₂)/(n₁+n₂). Weighted. Equal d: 2xy/(x+y). Ek figure galat copy.

Yahan kya nahi

CI 0–50, assumed mean, median formula, mode, ogive — Statistics. 5 log, ek 40 nikalta — Age. AP ka a+(n−1)d, Sₙ — Arithmetic progressions. 8:12:15 — Ratio. 25% of 80 — Percentage. Boats / 3 km east — TSD. CP–SP — Profit/loss.

Hamesha sum socho, average nahi jodna. Do classes ka average = averages ka average tabhi jab n barabar. Warna weighted. Equal time pe speed ka simple mean; equal distance pe 2xy/(x+y) — 40 aur 60 ka 50 nahi, 48.

2. Sum, list, pehle n

10, 20, 30. Sum 60, n=3, avg 20. Mean fulcrum hai — left 10 utna pull, right 30 utna.

10 20 30

10 aur 30 ka beech 20 — yahan teen numbers, AP formula Sₙ nahi. Coordinate midpoint alag chapter.

KyaAverage
Pehle n natural(n+1)/2   → n=20 ⇒ 10.5
Pehle n oddn   → pehle 10 odd ⇒ 10
Pehle n evenn+1   → pehle 10 even ⇒ 11

Har number +k ⇒ average +k. Har ×k ⇒ average ×k. 5, 15, 25, 35, 45 ka avg 25 (beech) — counting AP nahi, sirf list.

Example

10, 20, 30. Average?

Solution

  1. 60/3

    20. Median Statistics pe sort ke baad; yahan mean.

Answer20
Example

Pehle 20 natural numbers ka average?

Solution

  1. (n+1)/2

    21/2=10.5. Sum 20×21/2=210, 210/20 same. Number series “agla” nahi.

Answer10.5
Example

Pehle 10 odd numbers ka average?

Solution

  1. = n

    10. (1+19)/2=10. Pehle 10 even = 11.

Answer10

3. Add, remove, cricket, galat figure

Naya number aaye to naya sum = purana sum + x, naya n = n+1. Target average A chahiye: x = A(n+1) − purana sum. Cricket yahi — “ab kitna chahiye”.

5 innings avg 32, 6th se overall 40 chahiye: purana sum 160, naya total 6×40=240, 6th = 80. 40 mat likhna — 40 se upar banana padega kyunki pehle 32 pe kheench rahe.
40 32 32 32 32 32 80

Paanch 32. Dashed 40. 6th 80 taaki mean 40 ho. Statistics frequency-table nahi. Age “5 log 32, ek 40 nikalta” alag — wahan nikalna, yahan add.

Remove: 6 numbers avg 20, ek 14 nikaalo. Sum 120 − 14 = 106, baaki 5, avg 21.2. (Age wala 5 log / 40 saal yahan mat copy.)

Galat copy: 10 numbers avg 15, 12 ko 21 likh diya. Galat sum 150. Sahi sum 150 − 21 + 12 = 141. Sahi avg 14.1.

Example

5 innings avg 32. Overall 40 ke liye 6th kitni?

Solution

  1. Sums

    160. Target 240. 6th=80. 40 trap. 8 trap (40−32).

Answer80
Example

6 numbers avg 20. 14 nikaala. Baaki avg?

Solution

  1. 106/5

    21.2. 14 < 20 isliye avg upar. Family 32/40 Age pe.

Answer21.2
Example

10 numbers avg 15. 12 ko 21 padha. Sahi avg?

Solution

  1. −21+12

    150−9=141. 141/10=14.1.

Answer14.1

4. Combined, weighted, equal-distance speed

30 students avg 40 marks, 20 students avg 50. Total sum 30×40 + 20×50 = 2200. Kul 50. Combined 44. (40+50)/2=45 galat — n alag.

Combined = (n₁a₁ + n₂a₂) / (n₁+n₂). Weighted: subjects 80, 70, 90 weights 2, 3, 1 → (160+210+90)/6 = 460/6 = 76⅔.

Do speeds, equal distance

40 km/h se jaana, 60 se aana, same road. Time alag (zyada der slow pe). Average speed = total d / total t = 2xy/(x+y) = 2×40×60/100 = 48, arithmetic 50 nahi. Equal time hota to (40+60)/2=50. Boats current, 3 km east–north — TSD. Yahan sirf is shortcut ka ghar.

40 km/h 60 km/h same d → avg 48, not 50

Do equal roads. Harmonic: 2xy/(x+y). Teen equal d: 20, 30, 60 → 3 / (1/20+1/30+1/60) = 30.

Avg of avgs tabhi jab equal n. Cricket 80 vs “40 hi khelo”. Grouped mean assumed-a Statistics. Ratio chain 8:12:15 nahi. Profit% CP pe nahi.
Example

30 @ 40 marks, 20 @ 50. Combined?

Solution

  1. 2200/50

    44. 45 equal-n trap.

Answer44
Example

80, 70, 90 weights 2, 3, 1. Weighted avg?

Solution

  1. 460/6

    76⅔. Simple (80+70+90)/3=80 alag.

Answer76⅔
Example

40 km/h aur 60 km/h, equal distance. Avg speed?

Solution

  1. 2xy/(x+y)

    4800/100=48. 50 mat. Boats TSD.

Answer48 km/h

5. Sawal — basic se pro

Q1 · Basic

10, 20, 30. Average?

  1. 20
  2. 30
  3. 15
  4. 60

Solution

  1. 60/3

    20. D sum.

Answer20
Q2 · Basic

Pehle 20 natural. Average?

  1. 10.5
  2. 10
  3. 20
  4. 21

Solution

  1. (20+1)/2

    10.5.

Answer10.5
Q3 · Basic

Pehle 10 odd. Average?

  1. 10
  2. 11
  3. 9
  4. 19

Solution

  1. = n

    10. B even-wale.

Answer10
Q4 · Medium

5 innings 32. 6th se avg 40. 6th?

  1. 80
  2. 40
  3. 8
  4. 48

Solution

  1. 240−160

    80. Statistics table nahi.

Answer80
Q5 · Medium

30 @ 40, 20 @ 50. Combined?

  1. 44
  2. 45
  3. 40
  4. 50

Solution

  1. 2200/50

    44. B equal-n.

Answer44
Q6 · Medium

40 km/h aur 60, equal d. Avg speed?

  1. 48
  2. 50
  3. 100
  4. 24

Solution

  1. 2xy/(x+y)

    48. B arithmetic. Boats nahi.

Answer48
Q7 · Medium

10 avg 15. 12 as 21. Sahi avg?

  1. 14.1
  2. 15
  3. 13.2
  4. 16.8

Solution

  1. 141/10

    14.1.

Answer14.1
Q8 · Pro

6 avg 20. 14 nikaala. Baaki?

  1. 21.2
  2. 18
  3. 20
  4. 17.2

Solution

  1. 106/5

    21.2. Age 5-log-32 nahi.

Answer21.2
Q9 · Pro

80, 70, 90 weights 2, 3, 1?

  1. 76⅔
  2. 80
  3. 70
  4. 90

Solution

  1. 460/6

    76⅔. B unweighted.

Answer76⅔
Q10 · Pro

Class 0–50, median class n/2. Is chapter?

  1. Nahi — Statistics
  2. Haan, average = median
  3. Cricket 40
  4. Ogive yahan

Solution

  1. Grouped

    A. Yahan ungrouped / cricket.

AnswerA

Practice

Pehle khud, phir neeche kholo.

12345 678910

P1

Pehle 10 even. Average?

  1. 11
  2. 10
  3. 20
  4. 12

Solution

  1. n+1

    11.

Answer11
P2

8 innings 36. 9th = 54. Naya avg?

  1. 38
  2. 45
  3. 36
  4. 54

Solution

  1. 342/9

    38.

Answer38
P3

5 numbers avg 18. Har +4. Naya avg?

  1. 22
  2. 18
  3. 14
  4. 90

Solution

  1. +4

    22.

Answer22
P4

12 @ 18, 8 @ 12. Combined?

  1. 15.6
  2. 15
  3. 18
  4. 12

Solution

  1. 312/20

    15.6.

Answer15.6
P5

9 innings 25. 10th se avg 30. 10th?

  1. 75
  2. 30
  3. 5
  4. 55

Solution

  1. 300−225

    75.

Answer75
P6

Equal d: 20, 30, 60 km/h. Avg?

  1. 30
  2. 36⅔
  3. 110
  4. 40

Solution

  1. 3 / Σ(1/v)

    30. B arithmetic.

Answer30
P7

7 avg 12. Ek 26 add. Naya?

  1. 13.75
  2. 19
  3. 12
  4. 26

Solution

  1. 110/8

    13.75.

Answer13.75
P8

5, 15, 25, 35, 45. Average?

  1. 25
  2. 22
  3. 45
  4. 15

Solution

  1. 125/5

    25. AP Sₙ formula nahi.

Answer25
P9

A:B:C = 8:12:15. Is chapter?

  1. Nahi — Ratio
  2. Haan, avg 12
  3. Cricket 8
  4. Statistics 15

Solution

  1. Chain

    A. Wahan 8:12:15.

AnswerA
P10

CP 200 SP 250 profit%. Is chapter?

  1. Nahi — Profit / loss
  2. Haan, avg 225
  3. 25% yahan
  4. HCF 50

Solution

  1. CP base

    A. Agle chapter.

AnswerA

← Index · Ratio · Statistics

Agle topic: Profit, loss, discount.