Average

UTKALPRESS • ARITHMETIC SERIES

THE AVERAGE

A Complete Arithmetic Handbook

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AVERAGE

The average is one of the most important concepts in arithmetic. It helps us represent a group of values by a single meaningful number.

CONCEPT • FORMULA • SHORTCUT • PRACTICE

✓ Basic Average and Fundamental Formula
✓ Finding Total and Missing Values
✓ Addition, Removal and Replacement
✓ Combined and Weighted Average
✓ Frequency and Assumed Mean Method
✓ Average Speed and Age Problems
✓ Shortcuts and Exam Techniques
✓ Solved Practice Questions

MASTER CONCEPT

Average × Number of Observations = Total

This single relationship is the foundation of most average problems.

UTKALPRESS • AVERAGE

BASIC CONCEPTS

Understand the foundation before using shortcuts

1. What is Average?

The average of a group of numbers is the value obtained when the total of all the observations is divided equally among the number of observations.

FUNDAMENTAL FORMULA Average = Total Sum ÷ Number of Observations

Example: Find the average of 12, 18, 20, 25 and 30.

Sum = 12 + 18 + 20 + 25 + 30 = 105
Number of observations = 5
Average = 105 ÷ 5
Answer = 21

2. Three Most Important Formulae

Average = Sum ÷ Number

Sum = Average × Number

Number = Sum ÷ Average

EXAM TRICK
Whenever an average is given, immediately think: Average × Number = Total.

3. Average of Consecutive Numbers

For consecutive or equally spaced numbers, the average can be found by taking the mean of the first and last values.

Average = (First Number + Last Number) ÷ 2

Example: Find the average of 11, 12, 13, 14 and 15.

Average = (11 + 15) ÷ 2
= 26 ÷ 2
= 13

Middle Number Rule

For an odd number of equally spaced observations, the middle value is the average.

21, 23, 25, 27, 29
Middle value = 25
Therefore Average = 25

4. Finding a Missing Number

Required Total = Average × Number of Values

Missing Number = Required Total − Sum of Known Values

Example: Average of 5 numbers is 24. Four numbers are 10, 20, 25 and 30.

Required total = 24 × 5 = 120
Known total = 10 + 20 + 25 + 30 = 85
Missing number = 120 − 85
Answer = 35

5. Effect of Equal Change

If every observation is increased by the same amount, the average also increases by that amount.

New Average = Old Average ± Change

Average = 40. Every number is increased by 7.
New average = 40 + 7
= 47

UTKALPRESS • AVERAGE

ADVANCED METHODS

Addition • Removal • Replacement • Combined • Weighted Average

6. When a Number is Added

New Average = (nA + x) ÷ (n + 1)

Here n = original number of observations, A = original average and x = newly added value.

Example: Average of 5 numbers is 20. A new number 32 is added.

Original total = 5 × 20 = 100
New total = 100 + 32 = 132
New average = 132 ÷ 6
= 22

7. When a Number is Removed

New Average = (Original Total − Removed Value) ÷ Remaining Values

Example: Average of 6 numbers is 25. The number 40 is removed.

Original total = 6 × 25 = 150
Remaining total = 150 − 40 = 110
New average = 110 ÷ 5
= 22

8. Replacement Formula

When one value is replaced by another, the total changes only by the difference between the two values.

Change in Average = (New Value − Old Value) ÷ Number of Values

Example: Average of 10 numbers is 35. Number 20 is replaced by 50.

Change = 50 − 20 = 30
Change in average = 30 ÷ 10 = 3
New average = 35 + 3
= 38

9. Combined Average

When two or more groups are combined, calculate their individual totals before finding the combined average.

Combined Average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂)

Example: 20 students have average 60 and 30 students have average 70.

Combined average = (20×60 + 30×70) ÷ 50
= (1200 + 2100) ÷ 50
= 3300 ÷ 50
= 66

10. Weighted Average

When different values have different levels of importance, use weighted average.

Weighted Average = Σ(wx) ÷ Σw

Example: Marks 60, 70 and 80 have weights 2, 3 and 5.

Weighted average = (60×2 + 70×3 + 80×5) ÷ 10
= 730 ÷ 10
= 73

11. Frequency Method

Average = Σfx ÷ Σf

Value (x)	Frequency (f)	fx
5	2	10
10	3	30
15	5	75
Total	10	115

Average = 115 ÷ 10 = 11.5

UTKALPRESS • AVERAGE

SOLVED PRACTICE

Shortcuts • Applications • Examination Questions

12. Average Speed

Average Speed = Total Distance ÷ Total Time

The ordinary average of two speeds is valid only under particular conditions. For equal distances travelled at speeds a and b:

Average Speed = 2ab ÷ (a + b)

Example: A car travels equal distances at 40 km/h and 60 km/h.

Average speed = (2×40×60) ÷ (40+60)
= 4800 ÷ 100
= 48 km/h

13. Important Shortcuts

If every value increases by x, average increases by x.
If every value decreases by x, average decreases by x.
If every value is multiplied by k, average is also multiplied by k.
If every value is divided by k, average is also divided by k.
For equally spaced numbers, average = (First + Last) ÷ 2.
For combined groups, calculate totals first.
For weighted data, multiply each value by its weight.

14. Solved Practice Questions

1 Average of 15, 20, 25, 30 and 35?
Sum = 125
Average = 125 ÷ 5 = 25

2 Average of first 25 natural numbers?
Average = (25+1) ÷ 2 = 13

3 Average of 8 numbers is 24. Find total.
Total = 24 × 8 = 192

4 Average of 6 numbers is 18. Five have total 80.
Required total = 18×6 = 108
Missing number = 108−80 = 28

5 Average age of 10 students is 16.
Total age = 16×10 = 160 years

6 Average of 5 numbers is 30. Add 42.
Old total = 150
New total = 192
New average = 192÷6 = 32

7 Average of 7 numbers is 25. Remove 39.
Old total = 175
Remaining total = 136
New average = 136÷6 = 22⅔

8 Average of 10 numbers is 40. Replace 25 by 45.
Change = 20
Average increase = 20÷10 = 2
New average = 42

9 20 students average 60; 30 students average 70.
Combined average = (20×60 + 30×70)÷50
= 3300÷50 = 66

10 Weighted average of 60, 70, 80 with weights 2,3,5.
= (120+210+400)÷10
= 73

11 Equal distances at 30 km/h and 50 km/h.
Average speed = (2×30×50)÷80
= 37.5 km/h

12 Average age of 6 people is 25. After 5 years?
New average = 25+5 = 30 years

13 Average of numbers from 21 to 41.
Average = (21+41)÷2 = 31

14 Average of 12 numbers is 35. Add 7 to every number.
New average = 35+7 = 42

15 Average marks = 75 in 8 subjects.
Total marks = 75×8 = 600

15. Final Formula Card

Concept	Formula
Average	Total ÷ Number
Total	Average × Number
Consecutive Numbers	(First + Last) ÷ 2
Combined Average	(n₁A₁+n₂A₂) ÷ (n₁+n₂)
Weighted Average	Σwx ÷ Σw
Frequency Average	Σfx ÷ Σf
Average Speed	Total Distance ÷ Total Time

★ GOLDEN RULE OF AVERAGE ★

Whenever you see an average problem, think:

AVERAGE × NUMBER = TOTAL

Then adjust the total if something is added, removed or replaced.