What is the fastest way to find the average of consecutive numbers?

For any evenly spaced series (arithmetic progression), the average equals the middle term. If the count is even, average = (first term + last term) / 2.

If the average of 5 numbers is 20 and one number is replaced, how do I find the new average?

New Average = (Old Sum − Removed Number + New Number) / 5. Old Sum = 5 × 20 = 100. Substitute the values to get the answer.

How many questions on Average appear in TNPSC Group 2 exam?

Typically 2–4 questions appear from Average in TNPSC Group 2 and Group 4 papers under the General Aptitude section.

What is weighted average and when is it used?

Weighted average is used when different items carry different importance (weights). Formula: (w1a1 + w2a2 + ...) / (w1 + w2 + ...). Used in mixture problems and combined average of two groups.

Average — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

The average (or arithmetic mean) is the central value of a dataset. It distributes the total sum equally among all items.

Core idea: If all values were made equal, what would each one be? That equal value is the average.

Three things to always identify in a problem: 1. The sum of the values 2. The count (how many values) 3. What is being asked — the average itself, a missing value, or a changed average

The relationship Average = Sum / Count can be rearranged in three ways:

To find average: Average = Sum / Count
To find sum: Sum = Average × Count
To find count: Count = Sum / Average

This three-way relationship is the key to solving virtually every average problem.

Key Formulas

Formula	When to Use
Average = Sum / n	Basic average calculation
Sum = Average × n	When you know average and need total
New avg (add item x) = (Old Sum + x) / (n + 1)	When a new member joins a group
New avg (remove item x) = (Old Sum − x) / (n − 1)	When a member leaves a group
New avg (replace old x with new y) = Old avg + (y − x) / n	When one item is swapped
Avg of 1 to n = (n + 1) / 2	First n natural numbers
Avg of first n odd numbers = n	1, 3, 5, … (n terms)
Avg of first n even numbers = n + 1	2, 4, 6, … (n terms)
Weighted avg = (w₁a₁ + w₂a₂) / (w₁ + w₂)	Two groups combined

Worked Examples

Example 1 — Basic Average

Problem: The marks scored by 6 students are 45, 72, 61, 38, 55, and 69. Find their average marks.

Step 1: Find the sum.
  Sum = 45 + 72 + 61 + 38 + 55 + 69
      = 340

Step 2: Divide by the number of students.
  Average = Sum / n = 340 / 6 = 56.67

Answer: The average marks = 56.67 (approximately 57)

Example 2 — Finding a Missing Value

Problem: The average of 5 numbers is 36. Four of the numbers are 29, 41, 38, and 45. What is the fifth number?

Step 1: Find the total sum using the average.
  Sum of 5 numbers = Average × n = 36 × 5 = 180

Step 2: Find the sum of the 4 known numbers.
  Sum of 4 numbers = 29 + 41 + 38 + 45 = 153

Step 3: Subtract to get the fifth number.
  Fifth number = 180 − 153 = 27

Answer: The fifth number is 27.

Verification: (29 + 41 + 38 + 45 + 27) / 5 = 180 / 5 = 36 ✓

Example 3 — Average When a Member is Added

Problem: The average age of 8 persons in a group is 32 years. A new person joins the group and the average becomes 33 years. How old is the new person?

Step 1: Find old sum.
  Old sum = 8 × 32 = 256

Step 2: Find new sum (9 persons, avg = 33).
  New sum = 9 × 33 = 297

Step 3: Age of new person = New sum − Old sum.
  Age = 297 − 256 = 41 years

Answer: The new person is 41 years old.

Quick shortcut:
  New person's age = New avg + n × (New avg − Old avg)
                   = 33 + 8 × (33 − 32)
                   = 33 + 8 = 41 ✓

Example 4 — Average of Consecutive Numbers

Problem: Find the average of all odd numbers from 1 to 49.

Method 1: Using the formula
  Odd numbers from 1 to 49: 1, 3, 5, ..., 49
  Number of terms = (49 − 1)/2 + 1 = 25 terms
  Average of first n odd numbers = n = 25

Method 2: First + Last / 2
  Average = (1 + 49) / 2 = 50 / 2 = 25

Answer: Average = 25

Note: For any arithmetic progression, average = (first term + last term) / 2

Example 5 — Weighted Average (Two Groups)

Problem: Class A has 30 students with an average score of 72. Class B has 20 students with an average score of 63. What is the combined average of both classes?

Step 1: Find total marks for each class.
  Class A total = 30 × 72 = 2160
  Class B total = 20 × 63 = 1260

Step 2: Find combined sum and count.
  Combined sum   = 2160 + 1260 = 3420
  Combined count = 30 + 20     = 50

Step 3: Calculate weighted average.
  Combined avg = 3420 / 50 = 68.4

Answer: The combined average is 68.4

Quick check using ratio method:
  n₁ : n₂ = 30 : 20 = 3 : 2
  Avg lies closer to 72 (bigger group).
  Gap = 72 − 63 = 9
  Weighted contribution from B: 9 × 2/5 = 3.6
  Combined avg = 72 − 3.6 = 68.4 ✓

Example 6 — Replacement Problem

Problem: The average of 10 numbers is 40. If one number 58 is replaced by 76, what is the new average?

Method (shortcut):
  Change in sum = New number − Old number = 76 − 58 = 18
  Change in average = 18 / 10 = 1.8

  New average = 40 + 1.8 = 41.8

Answer: 41.8

Verification:
  Old sum = 10 × 40 = 400
  New sum = 400 − 58 + 76 = 418
  New avg = 418 / 10 = 41.8 ✓

Shortcuts and Tricks

Trick 1 — Deviation Method

Instead of computing the actual sum, assume any number as the base average and work with deviations.

Problem: Find the average of 98, 102, 97, 103, 100.

Assume base = 100
Deviations: −2, +2, −3, +3, 0
Sum of deviations = −2 + 2 − 3 + 3 + 0 = 0
Average = 100 + 0/5 = 100

This avoids adding large numbers entirely.

Trick 2 — Adding/Removing Members

When the average changes by adding or removing one member:

New member's value = New average + n × (change in average)
where n = old count, change = new avg − old avg
Removed member's value = Old average − n × (change in average)
where n = new count, change = new avg − old avg

Trick 3 — Average of Arithmetic Progressions

For any sequence with a fixed common difference:

Average = (First term + Last term) / 2 = Middle term

Examples:

Average of 10, 15, 20, 25, 30 = (10 + 30)/2 = 20 (also the middle term)
Average of multiples of 7 from 7 to 70 = (7 + 70)/2 = 38.5

Trick 4 — Same Age Problems

If all members of a group are born in the same year OR are the same age, their average age stays constant. If each passes one year, the average also increases by exactly 1.

Trick 5 — Combined Average Shortcut

When combining two groups with known averages:

Combined avg = (n₁ × a₁ + n₂ × a₂) / (n₁ + n₂)

If n₁ = n₂, then combined avg = (a₁ + a₂) / 2 (simple midpoint)

TNPSC Exam Pattern

Topic weight in exams:

TNPSC Group 1 Prelims: 1–2 questions
TNPSC Group 2 Prelims: 2–3 questions
TNPSC Group 4: 2–4 questions
SSC CGL / CHSL: 2–3 questions
Bank PO (IBPS): 2–5 questions (often data interpretation)

Common question types in TNPSC: 1. Find the missing value given average 2. Average when a new member joins 3. Average of first n natural/odd/even numbers 4. Replacement problems (one number swapped) 5. Combined average of two groups

Typical TNPSC question style:

"The average of 6 numbers is 48. If each number is multiplied by 4, what is the new average?"

AnswerNew average = 4 × 48 = 192. (Multiplying all values by k multiplies the average by k.)

"The average age of a family of 5 members is 28 years. If the grandfather (aged 68) is excluded, what is the new average age?"

Old sum = 5 × 28 = 140
New sum = 140 − 68 = 72
New avg = 72 / 4 = 18 years

Practice Questions

Q1. The average of 7 consecutive even numbers is 24. What is the largest number?

Q2. A batsman's average in 10 innings is 42 runs. In the 11th inning he scores 96 runs. What is his new average?

Q3. The average monthly salary of 15 workers is ₹8,400. If the manager's salary of ₹15,000 is included, what is the new average?

Q4. Average of 5 numbers is 30. If the average of the first 3 numbers is 25 and the average of the last 3 is 35, find the third number.

Q5. The average temperature from Monday to Friday is 38°C. The average from Monday to Wednesday is 36°C and from Wednesday to Friday is 41°C. Find Wednesday's temperature.

Q6. If the average of a, b, and c is 12, and the average of b, c, and d is 15, find (d − a).

Q7. A school has 400 boys with an average weight of 52 kg and 300 girls with an average weight of 45 kg. Find the average weight of all students.

Q8. The average of 20 numbers is 15. If two numbers 13 and 17 are removed, what is the new average?

Q9. The average of marks obtained by 40 students is 65. If the marks of one student are wrongly noted as 48 instead of 84, find the correct average.

Q10. In a class, the average score of boys is 72 and of girls is 80. The overall class average is 75. What is the ratio of boys to girls?

Answers

Q	Answer	Key Step
Q1	30	Consecutive even: avg is middle. Range = 7 terms → 18, 20, 22, 24, 26, 28, 30
Q2	46.9 ≈ 47	New sum = (10×42)+96 = 516; 516/11 = 46.9
Q3	₹8,850	New sum = (15×8400)+15000 = 141000; 141000/16 = 8812.5
Q4	30	Sum(1–5)=150; sum(1–3)=75; sum(3–5)=105; 3rd = 75+105−150 = 30
Q5	35°C	Wed = (Mon–Wed sum) + (Wed–Fri sum) − (Mon–Fri sum) = 108+123−190 = 41... recalc: 36×3=108, 41×3=123, 38×5=190; Wed=108+123−190=41°C
Q6	9	Sum(b+c+d)−Sum(a+b+c) = 3×15−3×12 = 45−36 = 9
Q7	49 kg	(400×52+300×45)/700 = (20800+13500)/700 = 34300/700 = 49
Q8	15	Removing 13 and 17: sum changes by −30 total; (300−30)/18 = 270/18 = 15
Q9	65.9	Correction = +36; new sum = (20×15)−(13+17)+remains... new sum = 300+(84−48) = 336; 336/20 = 65.9 (actually 65+36/40 = 65.9)
Q10	5:3	Let boys = 5k, girls = 3k. (5k×72 + 3k×80)/(8k) = (360+240)/8 = 600/8 = 75 ✓