Averages & Weighted Averages

Section A · Futures First International — Market Trading Intern OA

📌 What Has Been Asked (OA & Rounds)

Format	Typical Question Style
OA (MCQ)	"Average of 5 numbers is 40. If one number is replaced by 60, average becomes 44. Find the replaced number."
OA	"Average age of a class of 30 students is 14. When the teacher's age is included, average increases by 1. Teacher's age?"
OA	"Average weight of A, B, C is 60 kg. Average of A and B is 55 kg. Find C's weight."
Buzzer	"Average of first 10 natural numbers?" → instant: 5.5
Interview	"Average trade price calculation with varying lot sizes" — weighted average in disguise

🧠 Core Concepts

1. Basic Average

Average = Sum of all values / Number of values
Sum = Average × Count

2. Standard Results (Memorize)

Set	Average	Sum
First n natural numbers	(n+1)/2	n(n+1)/2
First n even numbers (2,4,...,2n)	n+1	n(n+1)
First n odd numbers (1,3,...,2n-1)	n	n²
Consecutive numbers from a to b	(a+b)/2	(a+b)/2 × count

3. Weighted Average

Weighted Avg = (w₁×x₁ + w₂×x₂ + ... + wₙ×xₙ) / (w₁ + w₂ + ... + wₙ)

4. Adding/Removing Elements

When a new value is added:

New average = (Old sum + New value) / (n + 1)

When a value is replaced (old → new):

New average = Old average + (New value − Old value) / n

When average changes by 'd' on adding one element to group of n:

New element = Old average + (n + 1) × d

⚡ Speed Tricks for Futures First

Trick 1: "New average after replacement"

"Average of 5 numbers is 40. One number replaced by 60, new average is 44."

Change in sum = 5 × (44 − 40) = 20
New value − Old value = 20
Old value = 60 − 20 = 40

Wait — that gives 40, meaning the sum increased by 20 so old value = 60 − 20 = 40

Trick 2: "Teacher/Boss added to group"

"30 students avg age 14. Adding teacher makes avg 15."

Teacher's age = New avg + n × (New avg − Old avg)
             = 15 + 30 × (15 − 14) = 15 + 30 = 45

Trick 3: Use deviations from assumed mean

To find average of: 78, 82, 85, 88, 77

Assume mean = 80
Deviations: −2, +2, +5, +8, −3 → Sum = +10
Average = 80 + 10/5 = 82

Trick 4: Average of an AP (Arithmetic Progression)

Average = middle term = (first + last)/2

Trick 5: Weighted average via alligation

Two groups with averages A₁ and A₂, and sizes n₁ and n₂:

Combined average lies between A₁ and A₂
n₁/n₂ = (A₂ − Combined) / (Combined − A₁)

This is just alligation applied to averages!

🎯 What Can Be Asked (Variations)

"Average marks of 30 students is 60. Average of passed students is 70, failed is 35. How many passed?" → Alligation: Passed:Failed = (60−35):(70−60) = 25:10 = 5:2. Total = 7 parts = 30 students. → Not exact. Let passed = x: 70x + 35(30−x) = 1800 → 35x = 750 → x ≈ 21.4 → check if question has nicer numbers.
"Average of 10 numbers is 15. Average of first 6 is 12. Average of last 5 is 18. Find the 6th number." → Sum all = 150. Sum first 6 = 72. Sum last 5 = 90. → 6th number is counted in both → 72 + 90 − 150 = 12
"A batsman's average after 20 innings is 45. After 21st innings, average becomes 47. Score in 21st innings?" → 47 × 21 − 45 × 20 = 987 − 900 = 87
"Avg weight of A,B,C,D is 60. Avg of A,B is 55 and C,D is 65. True or impossible?" → (55×2 + 65×2)/4 = 240/4 = 60 ✓ Consistent

📝 Practice Questions (with Answers)

Q1. Average of 8 numbers is 25. If each number is increased by 5, what is the new average?

A: 25 + 5 = 30 (adding constant to each → average increases by same constant)

Q2. The average age of a group of 12 friends is 28. If 2 friends of ages 24 and 22 leave, what is the new average?

A: Sum = 336. New sum = 336 − 24 − 22 = 290. New avg = 290/10 = 29

Q3. Average of first 50 natural numbers?

A: (50+1)/2 = 25.5

Q4. A class of 40 students has avg score 72. Another class of 60 students has avg score 83. Combined average?

A: (40×72 + 60×83)/100 = (2880 + 4980)/100 = 7860/100 = 78.6

Q5. Average of 6 numbers is 30. If one number is removed, average becomes 28. Find the removed number.

A: Sum = 180. New sum = 28 × 5 = 140. Removed = 180 − 140 = 40

Q6. A cricketer's average in 40 matches is 50. His avg in last 10 matches is 80. Average in first 30?

A: Sum all = 2000. Sum last 10 = 800. Sum first 30 = 1200. Avg = 1200/30 = 40

🔑 Exam Strategy

Averages are quick wins — most solvable in <15 seconds
The "teacher/new member added" pattern appears every year — formula: new_avg + n × change
Weighted averages are essentially alligation — use the cross method
Never compute full sums when you can use deviation from assumed mean
In trading context: "average trade price" = weighted average by lot size — same concept

06_averages.md