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

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Percentage 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.

Percentage: Complete Guide with Formulas, Conversions & Examples

What is Percentage?

Percentage is a way to express a number as a fraction of 100. The term \"percent\" comes from the Latin \"per centum,\" meaning \"by the hundred.\" It\'s a fundamental concept used in everyday life, business, mathematics, and competitive exams to compare quantities and express proportions.

Percentage = (Actual Value / Total Value) × 100

Example: If you score 45 marks out of 60, your percentage = (45/60)×100 = 75%

Basic Percentage Calculations

Finding Percentage of a Number

Percentage of a number = (Percentage/100) × Number

Example Problem: What is 30% of 250?

Convert 30% to decimal: 30/100 = 0.3
Multiply: 0.3 × 250 = 75
Alternatively: (30/100) × 250 = (30×250)/100 = 75

Finding Number When Percentage is Given

Number = (Given Value / Percentage) × 100

Example Problem: If 40% of a number is 20, find the number.

Let the number be x
40% of x = 20
(40/100) × x = 20
x = (20 × 100) / 40 = 50

The number is 50

Example 2: Which number is 50% less than 90?

50% of 90 = (50/100) × 90 = 45
50% less means subtract from original: 90 - 45 = 45
Alternatively: 50% less = 50% remains = 50% of 90 = 45

Percentage Increase and Decrease

Percentage Increase

% Increase = [(New Value - Original Value) / Original Value] × 100

Example: Population increased from 2,000 to 2,500

% Increase = [(2500-2000)/2000]×100 = (500/2000)×100 = 25%

Percentage Decrease

% Decrease = [(Original Value - New Value) / Original Value] × 100

Example: Profits decreased from ₹8,000 to ₹7,000

% Decrease = [(8000-7000)/8000]×100 = (1000/8000)×100 = 12.5%

Decimal to Fraction Conversion - Essential Percentages

Common Percentage to Fraction Conversions

Memorizing these conversions can significantly speed up percentage calculations:

Basic Conversions

Memorize these:
50% = ½ = 0.5
25% = ¼ = 0.25
75% = ¾ = 0.75
20% = ⅕ = 0.2
40% = ⅖ = 0.4
60% = ⅗ = 0.6
80% = ⅘ = 0.8

Important Thirds and Sixths

33.33% = ⅓ ≈ 0.333
66.66% = ⅔ ≈ 0.667
16.66% = ⅙ ≈ 0.1667
83.33% = ⅚ ≈ 0.8333

Example: 16.66% of 120 = ⅙ × 120 = 20

Eighth Conversions

12.5% = ⅛ = 0.125
37.5% = ⅜ = 0.375
62.5% = ⅝ = 0.625
87.5% = ⅞ = 0.875

Example: 37.5% of 160 = ⅜ × 160 = 60

Ninth Conversions

11.11% = ⅑ ≈ 0.111
22.22% = ²⁄₉ ≈ 0.222
33.33% = ⅓ (already covered)
44.44% = ⁴⁄₉ ≈ 0.444

Application Examples

Example 1: What is 50% of 108?

50% = ½, so ½ × 108 = 54

Example 2: What is 33.33% of 99?

33.33% = ⅓, so ⅓ × 99 = 33

Example 3: What is 12.5% of 200?

12.5% = ⅛, so ⅛ × 200 = 25

Example 4: What is 62.5% of 80?

62.5% = ⅝, so ⅝ × 80 = 50

Magic Cycles - Special Percentage Patterns

The Magic Circle of 1/7

When 1 is divided by 7, we get 0.142857142857... This repeating decimal has unique properties where multiplying it by integers 1-6 rearranges the same digits.

Magic 7 Conversions:
14.2857% = ¹⁄₇ ≈ 0.142857
28.5714% = ²⁄₇ ≈ 0.285714
42.8571% = ³⁄₇ ≈ 0.428571
57.1428% = ⁴⁄₇ ≈ 0.571428
71.4285% = ⁵⁄₇ ≈ 0.714285
85.7142% = ⁶⁄₇ ≈ 0.857142

Example: What is 28.5714% of 140?

28.5714% = ²⁄₇, so ²⁄₇ × 140 = 40

Other Important Magic Cycles

The 1/13 Pattern

7.6923% = ¹⁄₁₃ ≈ 0.076923
15.3846% = ²⁄₁₃ ≈ 0.153846
23.0769% = ³⁄₁₃ ≈ 0.230769
30.7692% = ⁴⁄₁₃ ≈ 0.307692

The 1/17 Pattern

5.8824% = ¹⁄₁₇ ≈ 0.058824
11.7647% = ²⁄₁₇ ≈ 0.117647
17.6471% = ³⁄₁₇ ≈ 0.176471

Quick Recognition Tips

Multiples of 7: 14.28%, 28.57%, 42.85%, etc.
Multiples of 9: 11.11%, 22.22%, 33.33%, etc.
Multiples of 11: 9.09%, 18.18%, 27.27%, etc.

Successive Percentage Change

Understanding Successive Changes

When multiple percentage changes are applied one after another to a quantity, we use successive percentage change formulas.

Two Successive Changes Formula

Net % Change = x + y + (x×y/100)
Where x and y are percentage changes
Use +ve for increase, -ve for decrease

Example 1: Price increases by 20%, then decreases by 10%

x = +20%, y = -10%
Net change = 20 + (-10) + [20×(-10)/100]
Net change = 20 - 10 - 2 = 8% increase
Verification: Start with 100 → 120 → 108 (8% increase from 100)

Example 2: Price increases by 10%, then decreases by 20%

x = +10%, y = -20%
Net change = 10 + (-20) + [10×(-20)/100]
Net change = 10 - 20 - 2 = -12% decrease

Example 3: Price increases by 20% each year for 2 years

x = +20%, y = +20%
Net change = 20 + 20 + [20×20/100]
Net change = 20 + 20 + 4 = 44% increase

Three Successive Changes

Method: Apply Formula Twice

First find net change for first two, then combine with third.

Example: Price changes by +10%, -20%, +30% successively

First combine +10% and -20%:

10 + (-20) + [10×(-20)/100] = 10 - 20 - 2 = -12%

Now combine -12% with +30%:

(-12) + 30 + [(-12)×30/100] = 18 - 3.6 = 14.4% increase

Verification: 100 → 110 → 88 → 114.4 (14.4% increase)

Percentage Change in Area, Volume, etc.

When sides increase by percentage

Rectangle/Length: If length increases by x% and breadth by y%
% change in area = x + y + (xy/100)

Cube/Side: If side increases by x%
% change in volume = 3x + 3x²/100 + x³/10000 ≈ 3x for small x

Example: Length increases by 20%, breadth by 10%

% change in area = 20 + 10 + (20×10/100) = 30 + 2 = 32% increase

Advanced Percentage Concepts

1. Percentage Error

% Error = |(True Value - Measured Value) / True Value| × 100

Example: True value = 100, Measured = 95

% Error = |(100-95)/100|×100 = 5%

2. Population Growth

Population after n years = P(1 + r/100)^n
Where P = initial population, r = growth rate

Example: Population 10,000 grows at 5% annually for 2 years

After 2 years = 10,000×(1.05)² = 11,025

3. Depreciation

Value after n years = P(1 - r/100)^n
Where P = initial value, r = depreciation rate

Example: Car worth ₹5,00,000 depreciates 10% annually

After 3 years = 5,00,000×(0.9)³ = ₹3,64,500

Percentage: Frequently Asked Questions

How to quickly calculate percentages mentally?

Memorize basic fractions: 50% = ½, 25% = ¼, 20% = ⅕, etc. For 15% of 80: 10% = 8, 5% = 4, so 15% = 12. For 37.5%: ⅜, so ⅜ × number.

Why is the magic cycle of 1/7 special?

When you multiply 0.142857 by 2, 3, 4, 5, or 6, you get the same digits rearranged: 0.285714, 0.428571, etc. This cyclic property makes calculations with sevenths easier.

How to handle three successive percentage changes?

Combine first two using the formula (x+y+xy/100), then combine the result with the third percentage change using the same formula.

What\'s the difference between percentage points and percent?

Percentage points refer to absolute difference: 10% to 15% is a 5 percentage point increase. Percent change refers to relative difference: 10% to 15% is a 50% increase.

How to reverse a percentage increase?

If price increased by 20% to become ₹120, original = 120/(1+20/100) = 120/1.2 = ₹100. Use formula: Original = New/(1±percentage/100).

Practice Problems

Problem 1

If a number is increased by 20% and then decreased by 20%, what is the net percentage change?

Solution: x=20, y=-20. Net change = 20-20+(20×-20/100)=0-4=-4%. Net decrease of 4%.

Problem 2

What is 37.5% of 256?

Solution: 37.5% = ⅜. ⅜ × 256 = 3 × 32 = 96.

Problem 3

A\'s salary is 20% less than B\'s. By what percent is B\'s salary more than A\'s?

Solution: Let B=100, A=80. B is more than A by (20/80)×100=25%.

Problem 4

Price increases by 25% but consumption decreases by 20%. What is the net effect on expenditure?

Solution: Expenditure = Price × Consumption. New expenditure = (1.25P)×(0.8C)=1.00PC. No change (0%).

Problem 5

If 71.4285% of 210 is x, find x.

Solution: 71.4285% = ⁵⁄₇. ⁵⁄₇ × 210 = 5 × 30 = 150.

Quick Reference Formulas

Basic Percentage:
Percentage = (Part/Whole) × 100
Part = (Percentage/100) × Whole
Whole = (Part/Percentage) × 100

Percentage Change:
% Change = [(New - Original)/Original] × 100
New = Original × (1 ± percentage/100)

Successive Changes:
Net % = x + y + (xy/100)
For three: First combine two, then with third

Reverse Calculation:
If increased by x% to become N: Original = N/(1+x/100)
If decreased by x% to become N: Original = N/(1-x/100)

Frequently Asked Questions

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

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

Yes, Percentage 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 Percentage Explained: Concepts, Formulas, and Easy Examples easily?

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

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

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