Averages Explained: Concepts, Formulas, and Easy Examples - Aptitude Questions & Answers

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Averages Explained: Concepts, Formulas, and Easy Examples is one of the most important topics in Quantitative Aptitude. In this lesson, you will learn concepts, formulas, shortcuts, solved examples, and aptitude questions with answers. This topic is useful for exams like SSC, Bank, CAT, TCS, and other competitive exams.

Average: Complete Guide with Formulas, Properties & Examples

What is Average?

Average (also called Arithmetic Mean) is a measure of central tendency that represents the typical value in a set of numbers. It\'s calculated by dividing the sum of all values by the number of values. Understanding averages is crucial for data analysis, statistics, and various competitive exams.

Average = Sum of Observations / Number of Observations
A = Sₙ / N

Example: If 3 apples weigh 100g, 150g, and 200g, average weight = (100+150+200)/3 = 450/3 = 150g

Basic Average Calculations

Finding Average

Example Problem: Find average weight of 4 boys with weights 30kg, 40kg, 50kg, 60kg

Sum of weights = 30 + 40 + 50 + 60 = 180 kg
Number of boys = 4
Average = 180/4 = 45 kg

Average weight = 45 kg

Example 2: Average of 5 numbers is 20. If four numbers are 15, 20, 25, 30, find the fifth number

Sum of 5 numbers = Average × Count = 20 × 5 = 100
Sum of first 4 numbers = 15 + 20 + 25 + 30 = 90
Fifth number = 100 - 90 = 10

Finding Sum When Average is Given

Sum of Observations = Average × Number of Observations
Sₙ = A × N

Example: Average of 8 numbers is 25. What is their sum?

Sum = 25 × 8 = 200

Finding Number of Observations

Number of Observations = Sum of Observations / Average
N = Sₙ / A

Example: Sum of numbers is 450, average is 30. How many numbers?

Count = 450/30 = 15 numbers

Properties of Average

Property 1: Adding/Subtracting Constant

If we add/subtract the same number to/from all observations, the average increases/decreases by that number.

If each observation is increased by x:
New Average = Old Average + x

If each observation is decreased by x:
New Average = Old Average - x

Example: Numbers: 10, 20, 30 (Average = 20)

Add 5 to each: 15, 25, 35 → Average = 25 (20 + 5)
Subtract 5 from each: 5, 15, 25 → Average = 15 (20 - 5)

Property 2: Multiplying/Dividing by Constant

If we multiply/divide all observations by the same number, the average gets multiplied/divided by that number.

If each observation is multiplied by x:
New Average = Old Average × x

If each observation is divided by x:
New Average = Old Average ÷ x

Example: Numbers: 2, 4, 6 (Average = 4)

Multiply by 3: 6, 12, 18 → Average = 12 (4 × 3)
Divide by 2: 1, 2, 3 → Average = 2 (4 ÷ 2)

Property 3: Average of Consecutive Numbers/AP

Odd Number of Terms in AP

Average = Middle term

Example: 1, 3, 5, 7, 9, 11, 13 (7 terms)

Middle term = 7, Average = (1+3+5+7+9+11+13)/7 = 49/7 = 7

Even Number of Terms in AP

Average = Average of two middle terms

Example: 2, 4, 6, 8, 10, 12 (6 terms)

Middle terms = 6, 8, Average = (6+8)/2 = 7

Check: Sum = 42, Count = 6, Average = 42/6 = 7

Consecutive Numbers Formula

Average of first n natural numbers = (n+1)/2
Average of n consecutive numbers = (First + Last)/2

Example: Average of first 10 natural numbers = (10+1)/2 = 5.5

Example: Average of 15, 16, 17, 18, 19 = (15+19)/2 = 17

Additional Properties of Average

Property 4: Combined Average

When two or more groups are combined, the overall average lies between the individual group averages.

Combined Average = (n₁A₁ + n₂A₂) / (n₁ + n₂)
Where:
n₁, n₂ = Number of observations in groups
A₁, A₂ = Averages of groups

Example: Group A: 20 students, average marks = 60
Group B: 30 students, average marks = 80
Find combined average

Total marks Group A = 20 × 60 = 1200
Total marks Group B = 30 × 80 = 2400
Total students = 20 + 30 = 50
Total marks = 1200 + 2400 = 3600
Combined average = 3600/50 = 72

Property 5: Weighted Average

When different observations have different weights/importance.

Weighted Average = Σ(Weight × Value) / ΣWeights

Example: Test 1: 80 marks (weight 2), Test 2: 90 marks (weight 3), Test 3: 70 marks (weight 1)

Weighted sum = (80×2) + (90×3) + (70×1) = 160 + 270 + 70 = 500
Total weight = 2 + 3 + 1 = 6
Weighted average = 500/6 = 83.33

Property 6: Effect of Removing/Adding an Observation

New Average = (Old Sum ± Value of added/removed observation) / (Old Count ± 1)

Example: Average of 5 numbers is 40. If one number 60 is added, new average becomes 42. Find original sum

Let original sum = S, original count = 5
Original average = S/5 = 40 → S = 200
New sum = 200 + 60 = 260
New count = 6
New average = 260/6 ≈ 43.33 (not 42)
Correct approach: (S + 60)/6 = 42 → S + 60 = 252 → S = 192

Types of Average Problems

Type 1: Missing Observation Problems

Example Problem: Average of 10 numbers is 35. If one number is excluded, average becomes 33. Find excluded number.

Sum of 10 numbers = 35 × 10 = 350
Sum of 9 numbers = 33 × 9 = 297
Excluded number = 350 - 297 = 53

Excluded number = 53

Example 2: Average of 15 numbers is 40. Two numbers 45 and 55 are removed. Find new average.

Sum of 15 numbers = 40 × 15 = 600
After removing: New sum = 600 - 45 - 55 = 500
New count = 15 - 2 = 13
New average = 500/13 ≈ 38.46

Type 2: Age Problems with Average

Example Problem: Average age of 5 family members is 30 years. If the youngest member (age 10) leaves, what is average age of remaining?

Total age of 5 = 30 × 5 = 150 years
After youngest leaves: Total age = 150 - 10 = 140 years
Remaining members = 4
New average = 140/4 = 35 years

Example 2: Average age of father, mother, and son is 35 years. If father is 40 and mother is 38, find son\'s age.

Total age of 3 = 35 × 3 = 105 years
Father + mother = 40 + 38 = 78 years
Son\'s age = 105 - 78 = 27 years

Type 3: Replacement/Substitution Problems

Example Problem: Average of 8 numbers is 25. If each number is multiplied by 2 and then 5 is added, find new average.

Using properties: Multiply by 2 → Average becomes 25×2 = 50
Add 5 to each → Average becomes 50 + 5 = 55

Verification: Original sum = 200, each number ×2+5: New sum = (200×2) + (8×5) = 400+40=440, New average=440/8=55

Type 4: Cricket Batting Average Problems

Example Problem: A batsman scores 60 runs in his 10th inning, increasing his average by 2 runs. Find his average after 10 innings.

Let average after 9 innings = x
Total runs after 9 innings = 9x
After 10th inning: Total = 9x + 60, Average = x + 2
Equation: (9x + 60)/10 = x + 2
9x + 60 = 10x + 20
x = 40 (average after 9 innings)
Average after 10 innings = 40 + 2 = 42

Type 5: Problems with Two Averages

Example Problem: In a class of 60 students, average marks of boys is 70 and girls is 80. If overall average is 74, find number of boys.

Let number of boys = B, girls = 60 - B
Total marks = 70B + 80(60 - B)
Overall average = [70B + 80(60 - B)]/60 = 74
70B + 4800 - 80B = 4440
-10B = -360
B = 36 boys

Alternative (Alligation): Boys:70, Girls:80, Overall:74

Ratio = (80-74):(74-70) = 6:4 = 3:2, Boys = (3/5)×60 = 36

Average: Frequently Asked Questions

What\'s the difference between average and weighted average?

Regular average treats all observations equally. Weighted average gives different importance (weights) to different observations. Example: Final grade where exams have different weights.

How to find average of consecutive numbers quickly?

For consecutive numbers, average = (first + last)/2. For first n natural numbers, average = (n+1)/2.

What happens when we remove the highest and lowest values?

Removing highest and lowest values eliminates extremes. The new average will be closer to the median and less affected by outliers.

How to handle average problems with age increase?

If ages of all members increase by 1 year each year, the average age also increases by 1 year. Total age increases by number of members each year.

Can average be less than all observations?

No, average always lies between the minimum and maximum values in the set. It cannot be less than the smallest or greater than the largest observation.

Practice Problems

Problem 1

Average of 7 numbers is 30. If each number is multiplied by 3 and then decreased by 5, find new average.

Solution: Using properties: 30×3 = 90, then 90-5 = 85. New average = 85.

Problem 2

Average of 25 observations is 36. If two observations 42 and 44 are incorrectly recorded as 24 and 46, find correct average.

Solution: Original sum = 25×36=900. Error: 42 recorded as 24 (-18), 44 as 46 (+2). Net error = -16. Correct sum=884. Correct average=884/25=35.36.

Problem 3

Average weight of 10 students is 40kg. If a teacher weighing 80kg joins, find new average weight.

Solution: Total weight=10×40=400kg. New total=400+80=480kg. New count=11. New average=480/11≈43.64kg.

Problem 4

Average of 3 numbers is 12. Average of first two is 9. Find third number.

Solution: Sum of 3=12×3=36. Sum of first two=9×2=18. Third number=36-18=18.

Problem 5

Find average of all even numbers from 1 to 50.

Solution: Even numbers: 2,4,...,50. Count=25. First=2, Last=50. Average=(2+50)/2=26.

Quick Formulas and Shortcuts

Basic Formula:
Average = Sum / Count
Sum = Average × Count
Count = Sum / Average

Consecutive Numbers:
First n natural numbers: (n+1)/2
First n even numbers: n+1
First n odd numbers: n
Numbers from a to b: (a+b)/2

Properties:
Adding x to each: New Avg = Old Avg + x
Multiplying by x: New Avg = Old Avg × x
Combined Avg = (n₁A₁ + n₂A₂)/(n₁+n₂)

Error Correction:
Correct Avg = (Wrong Sum ± Error)/Count
Where Error = Difference between correct and wrong values

Frequently Asked Questions

What is Averages Explained: Concepts, Formulas, and Easy Examples?

Averages Explained: Concepts, Formulas, and Easy Examples is an important aptitude topic used in competitive exams that tests your logical reasoning and problem-solving abilities.

Is Averages Explained: Concepts, Formulas, and Easy Examples important for competitive exams?

Yes, Averages Explained: Concepts, Formulas, and Easy Examples is frequently asked in SSC, Bank, CAT, TCS, and other placement exams. It's essential to master this topic for better scores.

How to prepare Averages Explained: Concepts, Formulas, and Easy Examples easily?

Practice solved examples, learn formulas and shortcuts, and attempt practice questions regularly to master Averages Explained: Concepts, Formulas, and Easy Examples.

What are the important formulas in Averages Explained: Concepts, Formulas, and Easy Examples?

Key formulas vary by topic, but generally include basic concepts, shortcuts, and standard problem-solving approaches specific to Averages Explained: Concepts, Formulas, and Easy Examples.

How many questions come from Averages Explained: Concepts, Formulas, and Easy Examples?

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