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📐 FRACTIONS, PERCENTAGES AND STANDARD FORM 📊

Complete Theory and Formulas for Cambridge IGCSE Grade 9 Mathematics

1️⃣ EQUIVALENT FRACTIONS

📌 Definition

Equivalent fractions are fractions that represent the same value but have different numerators and denominators. They are created by multiplying or dividing both the numerator and denominator by the same non-zero number.

Key Formula:

a b = a × k b × k

where k is any non-zero number

🔍 Cross Multiplication Method

To find a missing value in equivalent fractions, use cross multiplication:

If a b = c d

Then: a × d = b × c

Example: Find x if 2 5 = x 26

Solution:
Cross multiply: 2 × 26 = 5 × x
52 = 5x
x = 52 ÷ 5
x = 10.4 or 52 5

✂️ Simplifying Fractions

To simplify a fraction to its lowest terms, divide both numerator and denominator by their Greatest Common Factor (GCF).

Simplification Formula:

a b = a ÷ GCF b ÷ GCF

Example: Simplify 36 54

Solution:
GCF of 36 and 54 = 18
36 ÷ 18 54 ÷ 18 = 2 3

2️⃣ OPERATIONS ON FRACTIONS

✖️ Multiplying Fractions

Multiplication Formula:

a b × c d = a × c b × d

⚠️ IMPORTANT: The word "of" in mathematics means MULTIPLY!

Example: Calculate 3 4 × 2 7

Solution:
3 × 2 4 × 7 = 6 28 = 3 14 (simplified)

➗ Dividing Fractions

Division Formula:

a b ÷ c d = a b × d c

Flip the second fraction and multiply!

Example: Calculate 8 ÷ 1 3

Solution:
8 ÷ 1 3 = 8 × 3 1 = 24

➕➖ Adding and Subtracting Fractions

Addition/Subtraction Formula:

a b ± c d = ad ± bc bd

Or find a common denominator first

🔑 KEY RULE: You MUST have a common denominator before adding or subtracting!

Example: Calculate 1 6 + 3 8

Solution:
LCM of 6 and 8 = 24
1 6 = 4 24 and 3 8 = 9 24

4 24 + 9 24 = 13 24

🔄 Mixed Numbers and Improper Fractions

Converting Mixed to Improper:

a b c = (a × c) + b c

Example: Convert 2 1 2 to improper fraction

Solution:
2 1 2 = (2 × 2) + 1 2 = 5 2

3️⃣ PERCENTAGES

📊 What is a Percentage?

A percentage is a way of expressing a number as a fraction of 100. The symbol % means "per hundred" or "out of 100."

Percentage Definition:

x% = x 100 = x ÷ 100

🔄 Conversion Formulas

From	To	Formula
Fraction	Percentage	ab × 100 = Percentage
Percentage	Fraction	x100 (then simplify)
Decimal	Percentage	Decimal × 100 = Percentage
Percentage	Decimal	Percentage ÷ 100 = Decimal

Examples:
• 3 4 = (3 ÷ 4) × 100 = 75%
• 35% = 35 ÷ 100 = 0.35
• 0.625 × 100 = 62.5%
• 24% = 24 100 = 6 25

📈 Finding a Percentage of an Amount

Percentage of Amount Formula:

x% of A = x 100 × A

OR

(x ÷ 100) × A

Example: Find 24% of 93,800

Solution:
Method 1: 0.24 × 93,800 = 22,512
Method 2: 24 100 × 93,800 = 22,512

⬆️ Percentage Increase

Percentage Increase Formula:

New Amount = Original × (1 + x 100 )

OR

New Amount = Original × (1 + x%)

where x is the percentage increase

Example: There were 27,500 cases of illness. The next year there was a 9% increase. Find the new number.

Solution:
New = 27,500 × (1 + 0.09)
New = 27,500 × 1.09
New = 29,975 cases

⬇️ Percentage Decrease

Percentage Decrease Formula:

New Amount = Original × (1 - x 100 )

OR

New Amount = Original × (1 - x%)

where x is the percentage decrease

Example: A car decreases by 12% per year. If worth $3,875 now, what was it worth 1 year ago?

Solution:
$3,875 = Original × 0.88
Original = 3,875 ÷ 0.88
Original = $4,403.41

📊 Finding Percentage Change

Percentage Change Formula:

% Change = Change in Value Original Value × 100%

Change = New Value - Original Value

Example: Abdul's height was 160 cm, now it's 172 cm. Find percentage increase.

Solution:
Change = 172 - 160 = 12 cm
% Increase = 12 160 × 100% = 7.5%

🔄 Reverse Percentage (Finding Original)

Reverse Percentage Formula:

If New = Original × Multiplier
Then Original = New Multiplier

Example: Population increased by 1.6% to 3.352 million. Find original.

Solution:
Multiplier = 1 + 0.016 = 1.016
Original = 3.352 1.016 = 3.299 million

🔗 Successive Percentages

⚠️ CRITICAL: You CANNOT simply add successive percentages!

Successive Percentage Formula:

After multiple changes:
Final = Original × (1 ± x₁%) × (1 ± x₂%) × (1 ± x₃%) ...

Multiply the multipliers!

Example: Salary increases by 10%, then by 20%. Total increase?

Solution:
Let original = $100
After 10%: 100 × 1.10 = $110
After 20%: 110 × 1.20 = $132
Total increase = 132 - 100 = 32
Percentage = 32% (NOT 30%!)

Multiplier Method: 1.10 × 1.20 = 1.32 = 32% increase

4️⃣ STANDARD FORM (SCIENTIFIC NOTATION)

🔬 Definition

Standard form is a way of writing very large or very small numbers using powers of 10.

Standard Form Format:

a × 10^k

where:
• 1 ≤ a < 10 (a is between 1 and 10, not including 10)
• k is an integer (positive or negative)

➡️ Converting TO Standard Form

For LARGE numbers:
1. Move decimal point LEFT until after first digit
2. Count how many places moved
3. Power is POSITIVE (+)

For SMALL numbers:
1. Move decimal point RIGHT to after first non-zero digit
2. Count how many places moved
3. Power is NEGATIVE (-)

Examples:
• 3,500,000 → Decimal moves 6 left → 3.5 × 10⁶
• 0.000042 → Decimal moves 5 right → 4.2 × 10⁻⁵
• 678,000 → Decimal moves 5 left → 6.78 × 10⁵

⬅️ Converting FROM Standard Form

Conversion Rules:

• Positive power: Move decimal RIGHT
• Negative power: Move decimal LEFT
• The power tells you HOW MANY places to move

Examples:
• 3.7 × 10⁴ → Move 4 right → 37,000
• 5.2 × 10⁻³ → Move 3 left → 0.0052
• 8.45 × 10⁶ → Move 6 right → 8,450,000

✖️ Multiplying in Standard Form

Multiplication Rule:

(a × 10^m) × (b × 10ⁿ) = (a × b) × 10^(m+n)

Multiply numbers, ADD powers!

Example: (2 × 10⁵) × (3 × 10⁷)

Solution:
= (2 × 3) × 10⁽⁵⁺⁷⁾
= 6 × 10¹²

⚠️ REMEMBER: If result is not in proper form (a ≥ 10 or a < 1), adjust it!

Example: (5 × 10⁴) × (4 × 10³)

Solution:
= (5 × 4) × 10⁽⁴⁺³⁾
= 20 × 10⁷
= 2.0 × 10⁸ (adjusted to proper form)

➗ Dividing in Standard Form

Division Rule:

(a × 10^m) (b × 10ⁿ) = a b × 10^(m-n)

Divide numbers, SUBTRACT powers!

Example: (8 × 10⁹) ÷ (4 × 10⁵)

Solution:
= 8 4 × 10^(9-5)
= 2 × 10⁴
= 20,000

➕➖ Adding/Subtracting in Standard Form

Addition/Subtraction Rule:

Convert to SAME POWER first!
Then add/subtract the number parts
Keep the common power

Example: (3 × 10⁸) + (2 × 10⁷)

Solution:
Convert to same power:
3 × 10⁸ = 30 × 10⁷
Now: 30 × 10⁷ + 2 × 10⁷ = 32 × 10⁷
= 3.2 × 10⁸ (in proper form)

5️⃣ ESTIMATION

🎯 Purpose of Estimation

Estimation helps check if calculated answers are reasonable by using approximations that are easy to compute mentally.

Estimation Strategy:

Round to 1 significant figure
Use simple, friendly numbers
Calculate mentally
Compare with exact answer

📐 Estimation Rules

Rounding Guidelines:
• Round to 1 significant figure for quick estimates
• Use powers of 10 where possible
• For standard form: round the coefficient to 1 s.f.

Example: Estimate 312 × 489

Solution:
Round: 312 ≈ 300, 489 ≈ 500
Estimate: 300 × 500 = 150,000
In standard form: (3 × 10²) × (5 × 10²) = 15 × 10⁴ = 1.5 × 10⁵

Example: Estimate 9.3 7.6 × 5.9 0.95

Solution:
Round: 9.3 ≈ 9, 7.6 ≈ 8, 5.9 ≈ 6, 0.95 ≈ 1
Estimate: 9 8 × 6 1 ≈ 1 × 6 = 6

📋 QUICK REFERENCE FORMULAS

Topic	Key Formula	Remember
Equivalent Fractions	ab = a×kb×k	Multiply/divide both by same number
Multiply Fractions	ab × cd = acbd	Multiply tops, multiply bottoms
Divide Fractions	ab ÷ cd = ab × dc	Flip second, then multiply
Add/Subtract Fractions	Need common denominator first!	Don't forget this step
Fraction → %	ab × 100	Multiply by 100
% → Fraction	x100	Put over 100, simplify
% of Amount	x100 × A	Convert to decimal or fraction
% Increase	New = Original × (1 + x%)	ADD to 1
% Decrease	New = Original × (1 - x%)	SUBTRACT from 1
% Change	ChangeOriginal × 100%	Change = New - Original
Standard Form	a × 10k where 1 ≤ a < 10	a between 1 and 10
Multiply Std Form	(a × 10m)(b × 10n) = ab × 10m+n	ADD powers
Divide Std Form	(a × 10m) ÷ (b × 10n) = ab × 10m-n	SUBTRACT powers

💡 ESSENTIAL TIPS FOR SUCCESS

🎯 Top 10 Tips:

1. Always simplify fractions to lowest terms in final answers
2. Show your work step-by-step for partial credit
3. Check reasonableness by estimating before and after
4. Remember BODMAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction
5. Use decimals or fractions for percentages based on ease
6. Standard form: 1 ≤ a < 10 (NOT equal to 10!)
7. Convert mixed numbers to improper fractions for multiply/divide
8. The word "of" means multiply in mathematics
9. To divide by a fraction: flip it and multiply
10. Read the question: answer in the format requested

🚫 COMMON MISTAKES TO AVOID:

❌ Adding percentages in successive changes
❌ Forgetting to convert mixed numbers
❌ Rounding too early in calculations
❌ Not adjusting standard form properly
❌ Mixing up increase/decrease formulas
❌ Cancelling when adding/subtracting fractions
❌ Thinking negative powers mean negative numbers

📝 WORKED EXAMPLES

Example 1: Complex Fraction

Problem: Calculate (56 + 14) ÷ 18

Solution:
Step 1: Add fractions (LCM = 12)
56 = 1012, 14 = 312
1012 + 312 = 1312

Step 2: Divide by 18
1312 ÷ 18 = 1312 × 81 = 10412

Step 3: Simplify
10412 = 263 = 823

Example 2: Percentage Problem

Problem: 93,800 students took exam. 19% got A, 24% got B, 31% got C, 10% got D, 11% got E. Rest got U.
a) Find % who got U
b) Find fraction who got B (lowest terms)
c) How many got A?

Solutions:
(a) Total = 19 + 24 + 31 + 10 + 11 = 95%
Grade U = 100% - 95% = 5%

(b) 24% = 24100 = 625

(c) 19% of 93,800 = 0.19 × 93,800 = 17,822 students

Example 3: Standard Form Calculation

Problem: Calculate n = aba+b where a = 3 × 10⁸ and b = 2 × 10⁷

Solution:
Numerator: ab = (3 × 10⁸)(2 × 10⁷) = 6 × 10¹⁵

Denominator: Convert to same power
a = 30 × 10⁷, b = 2 × 10⁷
a + b = 32 × 10⁷ = 3.2 × 10⁸

Division: n = 6 × 10¹⁵3.2 × 10⁸ = 1.875 × 10⁷

To 3 s.f.: n = 1.88 × 10⁷

Example 4: Successive Percentages

Problem: Salary increases 10%, then 20%. Find total % increase.

Solution:
Let original = $100
After 10%: 100 × 1.10 = $110
After 20%: 110 × 1.20 = $132
Increase = 132 - 100 = $32
Percentage = 32100 × 100% = 32%

Quick Method: 1.10 × 1.20 = 1.32 → 32% increase

🎓 Good Luck with Your Studies! 🎓

Practice regularly, show your working, and check your answers!

Remember: Mathematics is about understanding patterns and relationships.
Take your time, be patient with yourself, and never stop asking questions!

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