Average – Complete Short Notes for Competitive Exams

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1. Definition

👉 Average (Mean) is the value that represents the central tendency of a group of numbers.
It tells us what is the “equal share” value if all quantities were equal.

$$\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$$

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2. Basic Formula

Type	Formula
Simple Average	$A = \frac{S}{N}$
Weighted Average	$A = \frac{w_1x_1 + w_2x_2 + ... + w_nx_n}{w_1 + w_2 + ... + w_n}$

Where:

• S = Sum of values

• N = Number of items

• w = weight

• x = value

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3. Common Conceptual Examples

Example 1:

Find average of 4, 6, 8, 10, 12

$$A = (4 + 6 + 8 + 10 + 12)/5 = 40/5 = 8$$

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Example 2:

Average age of 5 people = 25 years.
Total age = 25 × 5 = 125 years.

If one more person joins with age 35:
New average = (125 + 35) / 6 = 160 / 6 = 26.67 years

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4. Relation Between Sum, Average, and Number

$$\text{Sum} = \text{Average} × \text{Number of terms}$$ $$\text{Average} = \frac{\text{Sum}}{\text{Number of terms}}$$ $$\text{Number of terms} = \frac{\text{Sum}}{\text{Average}}$$

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5. Important Cases

Case 1: When one number changes

If average of n numbers is A, and one number x is replaced by y:

$$\text{New Average}=A+\frac{(y-x)}{\mathrm{n}}$$

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Case 2: When a new item is added

Old Average = A
New Average = A'
Number of items increases from n to (n + 1)

Then:

$$\text{New Value}=(n+1)A^{\mathrm{\prime }}-nA$$

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Case 3: When an item is removed

If one item x is removed:

$$x=nA-(n-1)A^{\mathrm{\prime }}$$

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6. Shortcut Tricks

✅ If the average of a set increases by d when a number x is added:
→ New number = Old Average + (d × number of old items)

✅ If average decreases by d when a number x is removed:
→ Removed number = Old Average − (d × number of remaining items)

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7. Combined Average (Two Groups)

If Group 1 → n₁ terms, avg = A₁
and Group 2 → n₂ terms, avg = A₂

Then combined average:

$$A=\frac{n_1{A}_1+n_2{A}_2}{n_1+n_{\mathrm{2<span\; class="vlist-s"></span>}}}$$

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Example:

Average age of 20 men = 30 years
Average age of 30 women = 25 years

$$A=\frac{20\times 30+30\times 25}{20+30}=\frac{600+750}{50}=27years$$

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8. Consecutive Numbers (Shortcuts)

Type	Average
1 to n	(n + 1)/2
Even numbers up to n	(n + 2)/2
Odd numbers up to n	(n + 1)/2

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Example:

Average of first 10 natural numbers = (10 + 1)/2 = 5.5
Average of first 10 even numbers = (10 + 2)/2 = 6

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9. Finding Missing Number

If average of 5 numbers = 10
Then total sum = 5 × 10 = 50
If 4 numbers are 8, 9, 10, 12 → sum = 39
Missing number = 50 − 39 = 11

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10. Average Speed

Average speed =

$$\frac{\text{Total Distance}}{\text{Total Time}}$$

If equal distances are covered at two speeds, S₁ and S₂:

$$\text{Average Speed}=\frac{2S_1{S}_2}{S_1+S_{\mathrm{2}}}$$

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Example:

A person travels 60 km at 30 km/hr and 60 km at 90 km/hr

$$\text{Avg speed}=\frac{2\times 30\times 90}{30+90}=\frac{5400}{120}=45km\mathrm{/}hr$$

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11. Average of Multiples

If numbers are in arithmetic progression (A.P.) like 3, 6, 9, 12...
Average = (First term + Last term)/2

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12. Weighted Average Concept

Used when quantities have different importance or weights.

Example	Formula
Exam marks (subject weights differ)	$\frac{Σ(w_i × x_i)}{Σw_i}$

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Example:

Math (40%) = 80 marks
Science (60%) = 70 marks
Weighted Average = (80×40 + 70×60)/100 = 74 marks

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13. Average in Percentage Change Problems

If averages increase or decrease successively by x% and y%:
Total % change = x + y + (xy)/100

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14. Practice Examples

Q1. Average of 8 numbers = 20.
Sum = 8 × 20 = 160.
If one number 24 added → new avg = (160 + 24)/9 = 20.44

Q2. Average of 10 numbers = 30.
One number 45 replaced by 25.
New average = 30 + (25 − 45)/10 = 28

Q3. Combined average of class A (avg 70, 20 students) and class B (avg 80, 30 students)
= (20×70 + 30×80)/50 = 76 marks

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15. Real-life Example

If average monthly expense of a person for 12 months = ₹8,000,
Total = 8,000 × 12 = ₹96,000 yearly expense.

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16. Quick Revision Summary

Concept	Formula
Simple Average	Sum / Count
Combined Average	(n₁A₁ + n₂A₂) / (n₁ + n₂)
Average Speed	2S₁S₂ / (S₁ + S₂)
Missing Item	Total − Known Sum
Replaced Item	A + (y − x)/n
Sum =	Average × No. of terms

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17. Short Tricks to Remember

🔹 Average of consecutive terms = middle term.
🔹 If equal numbers added/subtracted → average changes equally.
🔹 If all numbers multiplied/divided by k → average also × or ÷ by k.
🔹 For AP: Average = (First + Last)/2.
🔹 For two speeds: use harmonic mean formula (2ab / a+b).

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18. Common Pitfalls

⚠ Don’t confuse average speed with mean of speeds — use total distance/total time.
⚠ Ensure equal quantities when applying harmonic mean.
⚠ Check unit consistency (hours vs minutes).

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19. Practice at a Glance

Type	Formula	Example
Simple Avg	Sum/N	(10+20+30)/3=20
Combined Avg	(n₁A₁+n₂A₂)/(n₁+n₂)	70,80→76
Avg Speed	2S₁S₂/(S₁+S₂)	30,90→45
Replaced Item	A + (y−x)/n	25,45→28
Consecutive Nos.	(1st+last)/2	1–9→5

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✅ In One Line:

“Average = Equal share value — find sum, divide by count, and apply ratio logic smartly.”