Ch 6: Average (Ausat) | Maths

Dosto, Average (ausat) ka matlab hai "sabko barabar-barabar baant do to hisse mein kitna aayega". Formula sirf ek hai: Average = Total ÷ Kitne. Bas!

CET Group D mein isse 1–2 questions aate hain aur ye sabse aasan marks hain. Trick ye hai ki har sawal mein pehle SUM nikalo — average se sum, aur sum se naya average. 90% sawal isi ek soch se ban jaate hain.

Ye chapter chhota hai par ismein exam ke 5 fix patterns hain — nayi entry, replacement, galat entry, average speed aur weighted average. Sab detail mein niche! 💪

1. Base Formula (yahi sab kuch hai)

⭐ Average = Sabka Total (Sum) ÷ Kitne (n) Isko ulta karke: Sum = Average × n aur n = Sum ÷ Average
Solved Example: 2, 4, 6, 8, 10 ka average?
Step 1: Sum = 30, n = 5 → Step 2: Average = 30/5 = 6 ✓

📌 Sabse zaroori aadat Jo bhi sawal ho, pehla kaam: average ko SUM mein badlo (Sum = avg × n). Baaki sab uske baad khud-ba-khud ho jata hai.

2. Series ke Ready-made Averages (ratt lo!)

Series	Average	Example
Pehle n natural numbers (1,2,3…)	(n+1)/2	1–20 → 21/2 = 10.5
Pehle n even numbers (2,4,6…)	n + 1	Pehle 10 even → 11
Pehle n odd numbers (1,3,5…)	n	Pehle 10 odd → 10
Consecutive (lagatar) numbers	(pehla + aakhri)/2	24,26,28,30 → (24+30)/2 = 27
Pehle n multiples of k	k(n+1)/2	3 ke pehle 5 multiple → 3×6/2 = 9

📌 Sabse kaam ki baat Kisi bhi barabar antar wali series (AP) ka average = (pehla + aakhri) ÷ 2. Beech wale numbers jodne ki zaroorat hi nahi!

3. Nayi Entry / Koi Nikal Gaya

✏️ Solved Example — teacher ki umar 10 students ki average umar 15 saal hai. Teacher ko milane par average 16 ho jata hai. Teacher ki umar?
Step 1: Students ka sum = 15 × 10 = 150
Step 2: Naya sum (11 log) = 16 × 11 = 176
Step 3: Teacher = 176 − 150 = 26 saal ✓

⚡ Fast trick (5 second) Naya banda = naya average + (badhotri × purani count) = 16 + (1 × 10) = 26 ✓
Soch: teacher ko khud ka 16 chahiye, aur 10 logon ko 1-1 saal upar uthana hai.

4. Replacement (ek gaya, ek aaya) — ⭐ FORMULA

⭐ Naye ka weight = Purane ka weight + (n × average mein badlaav) Solved Example: 8 logon ka average weight 2.5 kg badh jata hai jab 65 kg wale ki jagah naya aadmi aata hai. Naye ka weight?
Step 1: Total badhotri = 8 × 2.5 = 20 kg
Step 2: Naya = 65 + 20 = 85 kg ✓

5. Galat Entry (wrong reading) ka Sudhaar

✏️ Solved Example 10 numbers ka average 42 nikla. Baad mein pata chala 56 ko galti se 26 padh liya tha. Sahi average?
Step 1: Sum mein kami = 56 − 26 = 30
Step 2: Average mein kami = 30 ÷ 10 = 3
Step 3: Sahi average = 42 + 3 = 45 ✓

6. Weighted Average (do group mile hue)

⭐ Weighted avg = (n₁×a₁ + n₂×a₂) / (n₁ + n₂) Solved Example: 30 ladkon ka average 16 saal, 20 ladkiyon ka 15 saal. Poori class ka average?
Step 1: Sum = (30 × 16) + (20 × 15) = 480 + 300 = 780
Step 2: Average = 780 ÷ 50 = 15.6 saal ✓ (dhyan do: 15.5 nahi — ladke zyada hain isliye unki taraf jhuka hai)

7. Average Speed (⚠️ sabse bada trap)

⭐ Average speed = Total distance ÷ Total time Agar doori barabar ho aur do speed x, y hon: Average speed = 2xy / (x + y)
Solved Example: 60 km/h se jaata hai, 40 km/h se lautta hai. Average speed?
Step 1: (2 × 60 × 40) / (60 + 40) = 4800/100 = 48 km/h ✓
⚠️ 50 km/h NAHI hota! Kyunki dheere wali speed par zyada time lagta hai.

✏️ Type 1 — Beech wala result nikalna 11 results ka average 50 hai. Pehle 6 ka 49, aakhri 6 ka 52. 6th result?
Step 1: Pehle 6 ka sum = 294, aakhri 6 ka sum = 312 → jodo = 606
Step 2: Ismein 6th result do baar gina gaya hai (dono taraf aata hai)
Step 3: Poora sum = 50 × 11 = 550 → 6th = 606 − 550 = 56 ✓

✏️ Type 2 — Ek number nikal do 5 numbers ka average 30 hai. Ek number 20 hata dene par baaki 4 ka average?
Step 1: Sum = 30 × 5 = 150
Step 2: Naya sum = 150 − 20 = 130 → average = 130/4 = 32.5 ✓

✏️ Type 3 — Cricket / batsman ka average Ek batsman ka 10 innings mein average 36 hai. 11th innings ke baad average 38 karna hai. Kitne run chahiye?
Step 1: Chahiye sum = 38 × 11 = 418, abhi ka sum = 36 × 10 = 360
Step 2: Runs = 418 − 360 = 58 ✓
Fast trick: naya average + (badhotri × purani innings) = 38 + (2 × 10) = 58

✏️ Type 4 — Consecutive numbers 4 lagatar even numbers ka average 27 hai. Sabse bada number?
Step 1: Numbers maano x, x+2, x+4, x+6 → average = x + 3
Step 2: x + 3 = 27 → x = 24 → numbers 24, 26, 28, 30 ✓

✏️ Type 5 — Umar (age) ka average Pati-patni ka average 25 saal tha shaadi ke waqt. 3 saal baad bacche ke saath teeno ka average 20 hai. Bacche ki umar?
Step 1: Shaadi par sum = 50. 3 saal baad dono = 50 + 6 = 56
Step 2: Teeno ka sum = 20 × 3 = 60 → bacha = 60 − 56 = 4 saal ✓

Trick 1: Pehle SUM banao, hamesha
Sum = Average × n
Average sunte hi use sum mein badal do. Average se seedha khelne mein galti hoti hai, sum mein jodna-ghatana ekdum aasan hai.

Trick 2: Nayi entry ka 5-second formula
Naya = naya average + (badlaav × purani count)
10 students avg 15 → teacher aane par 16: teacher = 16 + (1×10) = 26. Do baar sum nikalne ki zaroorat nahi!

Trick 3: Deviation (chhota-bada) method
Bade numbers ko "assume karke" ghatao
102, 104, 106, 108 ka average → 100 maan lo, bachte hain 2,4,6,8 (avg 5) → answer 100 + 5 = 105. Bade numbers jodne se bacho!

Trick 4: Lagatar (consecutive) series ka average
(pehla + aakhri) ÷ 2
Barabar antar wali kisi bhi list ka average — bas do sire jodo aur aadha karo. Beech ke 20 numbers ho to bhi ek hi step!

Trick 5: Average speed ka trap
Barabar doori → 2xy/(x+y)
60 & 40 → 48 (na ki 50). Aur 3 speeds hon to 3xyz/(xy+yz+zx). Yaad rakho: answer hamesha simple average se KAM aayega.

❌ Galti 1: Average speed mein dono speed ka simple average 60 aur 40 ka average speed 48 hai, 50 NAHI. Barabar doori par formula 2xy/(x+y) lagta hai — kyunki dheeri speed par zyada time kharch hota hai.

❌ Galti 2: Weighted average mein dono averages ka average 30 ladke (16 saal) + 20 ladkiyan (15 saal) ka class average 15.6 hai, 15.5 NAHI. Har group ko uski ginti (weight) se multiply karna zaroori hai.

❌ Galti 3: Replacement mein n se multiply na karna 8 logon ka average 2.5 kg badha to weight 2.5 kg nahi, 8 × 2.5 = 20 kg zyada hua. Naya banda = 65 + 20 = 85 kg.

❌ Galti 4: Galat entry mein plus-minus ulta karna Sahi number galat se bada tha (56 vs 26) → sum badhega → average badhega (42 → 45). Agar sahi number chhota hota to average ghatta.

❌ Galti 5: Beech wale result ko ek baar ginna "Pehle 6 + aakhri 6" mein 6th result do baar aa gaya (6+6 = 12, jabki total 11 hain). Isiliye 6th = (dono sums) − (total sum). Ye overlap dhyan mein rakho.

Average ka base formula?tap 👆

Sum ÷ n (aur Sum = avg × n)

Pehle n natural numbers ka average?tap 👆

(n+1)/2

Pehle n EVEN numbers ka average?tap 👆

n + 1

Pehle n ODD numbers ka average?tap 👆

n

Consecutive numbers ka average?tap 👆

(pehla + aakhri) / 2

1 se 20 tak ka average?tap 👆

10.5 (21/2)

Replacement mein naya weight?tap 👆

Purana + (n × average ka badlaav)

Nayi entry (banda joda) ka fast formula?tap 👆

Naya avg + (badlaav × purani count)

Weighted average ka formula?tap 👆

(n₁a₁ + n₂a₂) / (n₁ + n₂)

Average speed (general)?tap 👆

Total distance ÷ Total time

Barabar doori, speed x aur y → average?tap 👆

2xy / (x + y)

60 aur 40 km/h ka average speed?tap 👆

48 km/h (50 nahi!)

Galat entry sudhaar ka rule?tap 👆

Antar ÷ n, phir average mein jodo/ghatao

3 ke pehle 5 multiples ka average?tap 👆

9 (k(n+1)/2 = 3×6/2)

Sab numbers mein k joda jaye to average?tap 👆

Average bhi k badh jayega

Sab numbers ko k se multiply karein to?tap 👆

Average bhi k guna ho jayega

Average kis range mein hota hai?tap 👆

Sabse chhote aur sabse bade ke beech

Deviation trick kya hai?tap 👆

Bade numbers se ek base ghatao, avg nikaalo, base wapas jodo

Situation	Formula
Average	Sum ÷ n · Sum = avg × n
Pehle n natural	(n+1)/2
Pehle n even / odd	n+1 / n
Consecutive series	(pehla + aakhri)/2
Pehle n multiples of k	k(n+1)/2
Naya banda joda	Naya avg + (badlaav × purani count)
Replacement	Purana ± (n × badlaav)
Galat entry	Antar ÷ n se average sudharo
Weighted average	(n₁a₁ + n₂a₂)/(n₁+n₂)
Average speed	Total distance / total time
Barabar doori, 2 speeds	2xy/(x+y)
Sab mein k joda / k se guna	Average bhi +k / ×k

🎯 Score: 0 / 0 Total questions: 14

Practice (exam-style)

Q1. The average of 2, 4, 6, 8 and 10 is:

💡 Sum 30 ÷ 5 = 6 (ya consecutive trick: (2+10)/2 = 6).

Practice (exam-style)

Q2. The average of the first 20 natural numbers is:

💡 (n+1)/2 = 21/2 = 10.5.

Practice (exam-style)

Q3. The average of the first 10 even numbers is:

💡 Pehle n even ka average = n+1 = 11 (2 se 20 tak).

Practice (exam-style)

Q4. The average of 5 numbers is 30. If one number 20 is removed, the average of the remaining numbers is:

💡 Sum 150 − 20 = 130, 130/4 = 32.5.

Practice (exam-style)

Q5. The average age of 10 students is 15 years. When the teacher's age is included, the average becomes 16 years. The teacher's age is:

💡 176 − 150 = 26 (ya trick: 16 + 1×10).

Practice (exam-style)

Q6. The average weight of 8 persons increases by 2.5 kg when a new person replaces one weighing 65 kg. The weight of the new person is:

💡 65 + (8 × 2.5) = 65 + 20 = 85 kg.

Practice (exam-style)

Q7. A man travels from A to B at 60 km/h and returns at 40 km/h. His average speed for the whole journey is:

💡 2xy/(x+y) = 4800/100 = 48 km/h (50 wala trap!).

Practice (exam-style)

Q8. The average of 11 results is 50. If the average of the first 6 is 49 and that of the last 6 is 52, the 6th result is:

💡 294 + 312 − 550 = 56 (6th do baar gina gaya tha).

Practice (exam-style)

Q9. The average of 10 numbers was found to be 42. Later it was discovered that 56 was misread as 26. The correct average is:

💡 Antar 30 ÷ 10 = 3 → 42 + 3 = 45.

Practice (exam-style)

Q10. The average of four consecutive even numbers is 27. The largest of these numbers is:

💡 x+3 = 27 → x = 24 → 24, 26, 28, 30 → largest 30.

Practice (exam-style)

Q11. A batsman has an average of 36 runs in 10 innings. How many runs must he score in the 11th innings to raise his average to 38?

💡 418 − 360 = 58 (ya 38 + 2×10).

Practice (exam-style)

Q12. The average of three numbers is 20. If two of them are 15 and 25, the third number is:

💡 Sum 60 − (15+25) = 20.

Practice (exam-style)

Q13. In a class, 30 boys have an average age of 16 years and 20 girls have an average age of 15 years. The average age of the whole class is:

💡 (480 + 300)/50 = 15.6 (weighted — 15.5 wala trap!).

Practice (exam-style)

Q14. The average of the first five multiples of 3 is:

💡 3,6,9,12,15 ka sum 45 ÷ 5 = 9 (k(n+1)/2 = 3×6/2).

Average = Sum ÷ n — aur har sawal ka pehla step hai average ko sum banana (Sum = avg × n). Series ke ready answers yaad rakho: pehle n natural = (n+1)/2, even = n+1, odd = n, aur kisi bhi lagatar series ka = (pehla + aakhri)/2. Nayi entry ya replacement mein n × badlaav jodna mat bhoolna (8 log, 2.5 kg → 20 kg). Weighted average mein har group ko uski ginti se tolo (15.6, na ki 15.5). Aur sabse bada trap — average speed = 2xy/(x+y), 60 aur 40 ka jawab 48 hai, 50 nahi! 🚀