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Quick Summary
- What this covers: Every concept you need to feel confident with everyday math — whole numbers, fractions, decimals, and percentages — built from first principles.
- Why it matters: These four ideas are the language of money, contracts, measurement, opportunity, and risk. Mastering them is the highest-leverage non-technical skill an adult can acquire.
- Key insight: Fractions, decimals, and percentages are not three different topics. They are the same number wearing different costumes. Once you see that, everything collapses into clarity.
- Who this is for: Adults who quietly struggle with numbers, professionals tired of reaching for a calculator, parents helping children, students wanting a single reference, and anyone who was ever told they "weren't a math person."
- How to use it: Read it once cover-to-cover. Then bookmark it and use the conversion tables, formula sheets, and shortcuts as your permanent reference.
Prologue: The Receipt That Started Everything
Maya stared at the restaurant receipt. The bill was 847.50. The tip line was blank. Her three friends looked at her expectantly.
"Just split it four ways and add 18%," said Raj, already reaching for his phone calculator.
Maya froze. Not because she didn't have the money — but because she didn't know how. Fractions? Percentages? Decimals? They all blurred together into a fog of half-remembered school lessons and quiet shame.
She's not alone. And if you've ever felt that same tightness in your chest when numbers come up — this entire guide is written for you.
We're going to follow Maya's journey from number-phobic to number-confident. Along the way, you will master every concept you need: whole numbers, fractions, decimals, and the mighty percentage.
No jargon. No judgment. Just clarity.
Let's begin.
PART ONE — THE WORLD OF NUMBERS
Chapter 1: What Even Is a Number?
Before Maya could tackle that restaurant bill, she needed to go back to basics. Way back.
A number is simply a way to measure, count, or label something. That's it. It's humanity's oldest technology — older than the wheel, older than fire-making tools.
When a shepherd in ancient Mesopotamia needed to know if all his goats came home, he placed one pebble in a pouch for each goat that left in the morning. When they returned, he matched pebbles to goats. That was counting. That was the birth of numbers.
The Types of Numbers You'll Meet
Here's your map of the number universe:
| Type | What It Means | Examples |
|---|---|---|
| Natural Numbers | The counting numbers (no zero) | 1, 2, 3, 4, 5, ... |
| Whole Numbers | Natural numbers + zero | 0, 1, 2, 3, 4, ... |
| Integers | Whole numbers + negatives | ..., -3, -2, -1, 0, 1, 2, 3, ... |
| Rational Numbers | Any number expressible as a fraction | ½, 0.75, -3, 7, 0.333... |
| Irrational Numbers | Cannot be written as a simple fraction | π (3.14159...), √2 (1.41421...) |
| Real Numbers | All rational + irrational numbers | Everything on the number line |
Your takeaway: Every percentage, every fraction, every decimal you'll ever encounter in daily life is a rational number. Master rational numbers, and you master everyday math.
Chapter 2: The Number Line — Your Secret Weapon
Maya's tutor, Mr. Ade, drew a single horizontal line on a whiteboard.
"This," he said, "is the most powerful picture in all of mathematics."
←---|----|----|----|----|----|----|----|----|----|----|----|----|---→
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
Everything has a place on this line. Every fraction, every decimal, every percentage (once converted) — they all sit somewhere on this line.
Key Properties of the Number Line
- Numbers increase as you move right.
- Numbers decrease as you move left.
- Zero is the dividing point between positive and negative.
- Between any two numbers, there are infinitely many other numbers. (This is where fractions and decimals live.)
Maya looked at the space between 0 and 1.
"There's... stuff in there?"
"Infinite stuff," Mr. Ade smiled. "That's where the magic begins."
Chapter 3: Place Value — The Architecture of Every Number
Before we can manipulate numbers, you need to understand how they're built.
Every digit in a number has a place value — its position determines its worth.
Place Value Chart
| Millions | Hundred Thousands | Ten Thousands | Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|---|---|---|---|
| 1,000,000 | 100,000 | 10,000 | 1,000 | 100 | 10 | 1 | . | 0.1 | 0.01 | 0.001 |
Example: The number 3,847.56
| Digit | Place | Value |
|---|---|---|
| 3 | Thousands | 3,000 |
| 8 | Hundreds | 800 |
| 4 | Tens | 40 |
| 7 | Ones | 7 |
| 5 | Tenths | 0.5 |
| 6 | Hundredths | 0.06 |
Total: 3,000 + 800 + 40 + 7 + 0.5 + 0.06 = 3,847.56
Why this matters: When you understand place value, decimals stop being scary. The digit after the decimal point is just tenths, then hundredths, then thousandths. That's it. No mystery.
PART TWO — FRACTIONS: THE MISUNDERSTOOD HERO
Chapter 4: The Inciting Incident — When Maya Met Fractions
It was a Tuesday when Maya's world cracked open.
Her manager at the design firm sent an email: "We need to allocate ⅓ of the budget to marketing, ¼ to development, and the rest to operations. Can you calculate the operations portion and have it on my desk by 3 PM?"
Maya's palms went slick.
If you've ever stared at fractions and felt your brain go blank — you're about to have your breakthrough moment.
Chapter 5: What Is a Fraction, Really?
A fraction is nothing more than a division that hasn't been completed yet.
Numerator ← "How many parts you HAVE"
─────────────
Denominator ← "How many EQUAL parts the whole is divided into"
The Anatomy of a Fraction
¾ means:
- The whole has been divided into 4 equal parts
- You have 3 of those parts
That's it. No hidden complexity. No trick.
Visual Representation
Imagine a pizza cut into 4 equal slices:
┌─────────┐
│ ██ │ ██ │ ██ = Shaded (you have these)
│ ██ │ ██ │ ░░ = Unshaded (you don't)
├────┼────┤
│ ██ │ ░░ │
│ ██ │ ░░ │
└─────────┘
3 out of 4 slices = ¾
Chapter 6: Types of Fractions
Maya learned there are several kinds.
1. Proper Fractions
The numerator is smaller than the denominator.
| Fraction | Meaning |
|---|---|
| ½ | 1 out of 2 parts |
| ¾ | 3 out of 4 parts |
| ⅝ | 5 out of 8 parts |
| 2/7 | 2 out of 7 parts |
Always less than 1 whole.
2. Improper Fractions
The numerator is equal to or larger than the denominator.
| Fraction | Meaning |
|---|---|
| 5/4 | 5 out of 4 parts (more than 1 whole) |
| 7/3 | 7 out of 3 parts |
| 9/9 | 9 out of 9 parts (exactly 1 whole) |
| 12/5 | 12 out of 5 parts |
Equal to or greater than 1 whole.
3. Mixed Numbers
A whole number combined with a proper fraction.
| Mixed Number | Meaning | Improper Equivalent |
|---|---|---|
| 1 ¼ | 1 whole and ¼ more | 5/4 |
| 2 ⅓ | 2 wholes and ⅓ more | 7/3 |
| 3 ⅘ | 3 wholes and ⅘ more | 19/5 |
Converting Between Mixed Numbers and Improper Fractions
Mixed → Improper
Formula: (Whole × Denominator + Numerator) / Denominator
Example: Convert 3 ⅖ to an improper fraction.
Step 1: Multiply whole number by denominator → 3 × 5 = 15
Step 2: Add the numerator → 15 + 2 = 17
Step 3: Place over the same denominator → 17/5
Result: 3 ⅖ = 17/5
Improper → Mixed
Formula: Divide numerator by denominator. Quotient = whole number; remainder = new numerator.
Example: Convert 17/5 to a mixed number.
Step 1: 17 ÷ 5 = 3 remainder 2
Step 2: Whole = 3, Remainder = 2, Denominator stays = 5
Result: 17/5 = 3 ⅖
Chapter 7: Equivalent Fractions — The Shape-Shifters
This was Maya's first "aha!" moment.
"Wait," she said. "½ and 2/4 and 3/6 and 50/100 are ALL the same thing?"
"Exactly," said Mr. Ade.
The Golden Rule of Equivalent Fractions
Whatever you do to the numerator, you must do to the denominator.
1 1 × 2 2 2 × 3 6 6 × 10 60
─── = ─────── = ─── = ─────── = ─── = ────────── = ────
2 2 × 2 4 4 × 3 12 12 × 10 120
All of these equal ½. They look different, but they represent the same amount.
Equivalent Fractions Table
| Simplest Form | ×2 | ×3 | ×4 | ×5 | ×10 |
|---|---|---|---|---|---|
| ½ | 2/4 | 3/6 | 4/8 | 5/10 | 10/20 |
| ⅓ | 2/6 | 3/9 | 4/12 | 5/15 | 10/30 |
| ¼ | 2/8 | 3/12 | 4/16 | 5/20 | 10/40 |
| ⅕ | 2/10 | 3/15 | 4/20 | 5/25 | 10/50 |
| ⅙ | 2/12 | 3/18 | 4/24 | 5/30 | 10/60 |
Chapter 8: Simplifying Fractions
The reverse skill — making fractions as small as possible.
Find the Greatest Common Factor (GCF) of numerator and denominator, then divide both by it.
Example: Simplify 18/24.
Step 1: Factors of 18 → 1, 2, 3, 6, 9, 18
Step 2: Factors of 24 → 1, 2, 3, 4, 6, 8, 12, 24
Step 3: GCF = 6
Step 4: 18 ÷ 6 = 3, 24 ÷ 6 = 4
Result: 18/24 = ¾
Quick GCF Reference
| Number Pair | GCF | Simplified |
|---|---|---|
| 4/8 | 4 | ½ |
| 6/9 | 3 | ⅔ |
| 10/25 | 5 | ⅖ |
| 12/16 | 4 | ¾ |
| 15/45 | 15 | ⅓ |
| 20/100 | 20 | ⅕ |
| 36/48 | 12 | ¾ |
Chapter 9: Comparing Fractions
Maya needed to know: Is ⅜ bigger or smaller than ⅖?
Method 1: Cross-Multiplication (Fast & Reliable)
Compare a/b and c/d by computing a × d vs. c × b.
Example: Compare ⅜ and ⅖.
3 × 5 = 15 vs. 2 × 8 = 16
15 < 16
Therefore: ⅜ < ⅖
Method 2: Common Denominator
⅜ = 15/40 (multiply top and bottom by 5)
⅖ = 16/40 (multiply top and bottom by 8)
15/40 < 16/40 → ⅜ < ⅖ ✓
Method 3: Convert to Decimals
⅜ = 3 ÷ 8 = 0.375
⅖ = 2 ÷ 5 = 0.400
0.375 < 0.400 → ⅜ < ⅖ ✓
Common Fractions Ranked Smallest to Largest
1/10 < ⅛ < 1/7 < ⅙ < ⅕ < ¼ < 2/7 < ⅓ < ⅜ < ⅖
< 3/7 < ½ < 4/7 < ⅗ < ⅝ < ⅔ < 5/7 < ¾
< ⅘ < ⅚ < 7/8 < 9/10
Chapter 10: Operations with Fractions
This is where Maya levelled up.
Addition of Fractions
Same denominator (easy):
a/c + b/c = (a + b)/c
2/7 + 3/7 = (2+3)/7 = 5/7
Different denominators (find common denominator):
Find the Least Common Denominator (LCD)Convert each fractionAdd the numeratorsSimplify if possible
Example: ⅔ + ¾
Step 1: LCD of 3 and 4 = 12
Step 2: ⅔ = 8/12 (multiply by 4/4)
¾ = 9/12 (multiply by 3/3)
Step 3: 8/12 + 9/12 = 17/12
Step 4: 17/12 = 1 5/12
Result: ⅔ + ¾ = 1 5/12
Subtraction of Fractions
Same process, but subtract.
Example: ⅚ - ⅜
Step 1: LCD of 6 and 8 = 24
Step 2: ⅚ = 20/24 (multiply by 4/4)
⅜ = 9/24 (multiply by 3/3)
Step 3: 20/24 - 9/24 = 11/24
Result: ⅚ - ⅜ = 11/24
Multiplication of Fractions
The easiest operation. Multiply straight across.
a/b × c/d = (a × c) / (b × d)
Example: ⅔ × ⅘
Numerators: 2 × 4 = 8
Denominators: 3 × 5 = 15
Result: ⅔ × ⅘ = 8/15
Pro tip: Simplify before multiplying to keep numbers small.
Example: 3/8 × 4/9
Cross-simplify first:
3 and 9 share factor 3 → simplify to 1 and 3
4 and 8 share factor 4 → simplify to 1 and 2
Now multiply: 1/2 × 1/3 = 1/6
Result: 3/8 × 4/9 = 1/6 (much cleaner than computing 12/72 and reducing).
Division of Fractions
Keep, Change, Flip (KCF) — the most memorable rule in fraction math.
a/b ÷ c/d = a/b × d/c
Example: ¾ ÷ ⅝
Step 1: Keep the first fraction → ¾
Step 2: Change ÷ to ×
Step 3: Flip the second fraction → 8/5
¾ × 8/5 = 24/20 = 6/5 = 1 ⅕
Result: ¾ ÷ ⅝ = 1 ⅕
Complete Operations Summary
| Operation | Rule | Example | Result |
|---|---|---|---|
| Add (same denom) | a/c + b/c = (a+b)/c | 2/5 + 1/5 | 3/5 |
| Add (diff denom) | Find LCD, convert, add | ½ + ⅓ = 3/6 + 2/6 | 5/6 |
| Subtract (same) | a/c - b/c = (a-b)/c | 5/8 - 3/8 | 2/8 = ¼ |
| Subtract (diff) | Find LCD, convert, subtract | ¾ - ⅔ = 9/12 - 8/12 | 1/12 |
| Multiply | a/b × c/d = ac/bd | ⅔ × ¾ | 6/12 = ½ |
| Divide | a/b ÷ c/d = a/b × d/c | ½ ÷ ¼ = ½ × 4/1 | 4/2 = 2 |
Chapter 11: Maya Solves the Budget Problem
Remember Maya's task? Allocate ⅓ to marketing, ¼ to development, the rest to operations.
Watch her solve it:
Step 1: Add the known fractions
⅓ + ¼
Step 2: Find LCD of 3 and 4 = 12
⅓ = 4/12
¼ = 3/12
Step 3: Add
4/12 + 3/12 = 7/12
Step 4: Subtract from the whole (12/12)
12/12 - 7/12 = 5/12
Operations gets 5/12 of the budget.
If the total budget is 120,000 (in any currency):
| Department | Fraction | Calculation | Amount |
|---|---|---|---|
| Marketing | ⅓ | 120,000 × ⅓ | 40,000 |
| Development | ¼ | 120,000 × ¼ | 30,000 |
| Operations | 5/12 | 120,000 × 5/12 | 50,000 |
| Total | 12/12 | 120,000 |
Maya sent the email at 2:47 PM. Thirteen minutes to spare.
She smiled for the first time in years at a math problem.
PART THREE — DECIMALS: FRACTIONS IN DISGUISE
Chapter 12: The Struggle — Decimals Everywhere
Two weeks into her journey, Maya noticed something: decimals were stalking her.
- Her bank balance: 2,341.87
- The fuel price: 1.459 per litre
- Her fitness app: "You ran 5.3 km today!"
- The weather: "Temperature: 22.5°C"
"Are these... fractions?" she asked Mr. Ade.
"Every single one of them," he said.
Chapter 13: Understanding Decimals
A decimal is simply a fraction whose denominator is a power of 10.
The decimal point separates the whole number part from the fractional part.
3 . 1 4 1 5 9
│ │ │ │ │ │ │
│ │ │ │ │ │ └── Hundred-thousandths (1/100,000)
│ │ │ │ │ └────── Ten-thousandths (1/10,000)
│ │ │ │ └────────── Thousandths (1/1,000)
│ │ │ └────────────── Hundredths (1/100)
│ │ └────────────────── Tenths (1/10)
│ └───────────────────── Decimal Point
└──────────────────────── Ones (1)
Decimal Place Value Chart
| Place Name | Value | Fraction | Example in 47.8362 |
|---|---|---|---|
| Tens | 10 | — | 4 (= 40) |
| Ones | 1 | — | 7 (= 7) |
| . (Decimal Point) | — | — | . |
| Tenths | 0.1 | 1/10 | 8 (= 0.8) |
| Hundredths | 0.01 | 1/100 | 3 (= 0.03) |
| Thousandths | 0.001 | 1/1000 | 6 (= 0.006) |
| Ten-thousandths | 0.0001 | 1/10000 | 2 (= 0.0002) |
47.8362 = 40 + 7 + 0.8 + 0.03 + 0.006 + 0.0002
Chapter 14: Converting Between Fractions and Decimals
Fraction → Decimal
Divide the numerator by the denominator.
| Fraction | Division | Decimal |
|---|---|---|
| ½ | 1 ÷ 2 | 0.5 |
| ¼ | 1 ÷ 4 | 0.25 |
| ⅓ | 1 ÷ 3 | 0.333... (repeating) |
| ⅕ | 1 ÷ 5 | 0.2 |
| ⅛ | 1 ÷ 8 | 0.125 |
| ⅙ | 1 ÷ 6 | 0.1666... (repeating) |
| 1/7 | 1 ÷ 7 | 0.142857... (repeating) |
| 1/9 | 1 ÷ 9 | 0.111... (repeating) |
| 1/11 | 1 ÷ 11 | 0.0909... (repeating) |
Decimal → Fraction
Read the decimal aloud, write it as a fraction, then simplify.
Example: Convert 0.75 to a fraction.
Step 1: 0.75 = "seventy-five hundredths" = 75/100
Step 2: Simplify → GCF of 75 and 100 = 25
Step 3: 75 ÷ 25 = 3, 100 ÷ 25 = 4
Result: 0.75 = ¾
Terminating vs. Repeating Decimals
Terminating decimals end after a finite number of digits:
½ = 0.5 ¼ = 0.25 ⅛ = 0.125
Repeating decimals have a pattern that goes on forever:
⅓ = 0.333... written as 0.3̄
⅙ = 0.1666... written as 0.16̄
1/7 = 0.142857... written as 0.142857̄
The rule: If the denominator (in simplest form) has only factors of 2 and/or 5, the decimal terminates. Otherwise, it repeats.
| Denominator | Prime Factors | Terminates? |
|---|---|---|
| 2 | 2 | ✅ Yes |
| 4 | 2² | ✅ Yes |
| 5 | 5 | ✅ Yes |
| 8 | 2³ | ✅ Yes |
| 10 | 2 × 5 | ✅ Yes |
| 20 | 2² × 5 | ✅ Yes |
| 25 | 5² | ✅ Yes |
| 3 | 3 | ❌ Repeats |
| 6 | 2 × 3 | ❌ Repeats |
| 7 | 7 | ❌ Repeats |
| 9 | 3² | ❌ Repeats |
| 11 | 11 | ❌ Repeats |
Chapter 15: Decimal Operations
Addition & Subtraction
Golden Rule: Line up the decimal points.
Example: 23.456 + 8.9
23.456
+ 8.900 ← Add trailing zeros to align
────────
32.356
Example: 50.2 - 13.875
50.200 ← Add trailing zeros
- 13.875
────────
36.325
Multiplication
Multiply as if there's no decimal point, then count total decimal places.
Example: 3.14 × 2.5
Step 1: Multiply 314 × 25 = 7,850
Step 2: Count decimal places: 3.14 has 2, 2.5 has 1 → Total = 3
Step 3: Place decimal 3 places from right: 7.850
Result: 3.14 × 2.5 = 7.85
Division
Move the decimal in the divisor to make it a whole number, then move it the same number of places in the dividend.
Example: 8.64 ÷ 3.2
Step 1: 3.2 → move decimal 1 place right → 32
Step 2: 8.64 → move decimal 1 place right → 86.4
Step 3: 86.4 ÷ 32 = 2.7
Result: 8.64 ÷ 3.2 = 2.7
Decimal Operations Summary
| Operation | Key Rule | Example | Result |
|---|---|---|---|
| Addition | Align decimal points | 4.56 + 2.3 | 6.86 |
| Subtraction | Align decimal points | 10.5 - 3.72 | 6.78 |
| Multiplication | Count total decimal places | 1.2 × 0.3 | 0.36 |
| Division | Make divisor a whole number | 6.5 ÷ 0.5 | 13 |
Chapter 16: Rounding Decimals
Sometimes you don't need all those decimal places.
Rule: Look at the digit ONE PLACE to the right of where you want to round.If it's 5 or more → round UPIf it's less than 5 → round DOWN
Example: Round 3.4567 to various places.
| Round to | Look at | Result |
|---|---|---|
| Whole number | 4 (tenths place) | 3 |
| 1 decimal place | 5 (hundredths place) | 3.5 |
| 2 decimal places | 6 (thousandths place) | 3.46 |
| 3 decimal places | 7 (ten-thousandths place) | 3.457 |
PART FOUR — PERCENTAGES: THE UNIVERSAL LANGUAGE OF COMPARISON
Chapter 17: The Transformation — Maya Discovers Percentages
It was the first of the month. Maya logged into her banking app and saw:
Savings Interest Rate: 4.5% per annum
Credit Card APR: 19.9%
Investment Return (Last Year): +12.3%
Grocery spending vs. last month: ↑ 8%
Every single insight her bank gave her was in percentages. Not fractions. Not decimals. Percentages.
"Why?" she asked Mr. Ade.
"Because percentage is the universal language of comparison. When everything is out of 100, you can compare anything to anything."
That single sentence changed Maya's relationship with numbers forever.
Chapter 18: What Is a Percentage?
Percent literally means "per hundred" (from Latin: per centum).
A percentage is a fraction with a denominator of 100.
25% = 25/100 = 0.25 = ¼
The Holy Trinity: Fraction ↔ Decimal ↔ Percentage
These three are the same number in different costumes.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/100 | 0.01 | 1% |
| 1/10 | 0.1 | 10% |
| 1/5 | 0.2 | 20% |
| 1/4 | 0.25 | 25% |
| 1/3 | 0.333... | 33.33...% |
| 1/2 | 0.5 | 50% |
| 3/5 | 0.6 | 60% |
| 2/3 | 0.666... | 66.67% |
| 3/4 | 0.75 | 75% |
| 4/5 | 0.8 | 80% |
| 9/10 | 0.9 | 90% |
| 1/1 | 1.0 | 100% |
| 5/4 | 1.25 | 125% |
| 3/2 | 1.5 | 150% |
| 2/1 | 2.0 | 200% |
Chapter 19: The Conversion Triangle
This is the single most important skill in practical mathematics.
FRACTION ──divide──→ DECIMAL ──× 100──→ PERCENTAGE
↑ │
└──────────────── ÷ 100 ─────────────────┘
Step-by-Step Conversions
Fraction → Decimal: Divide numerator by denominator.
¾ → 3 ÷ 4 = 0.75
Decimal → Percentage: Multiply by 100 (move decimal 2 places right).
0.75 → 0.75 × 100 = 75%
Percentage → Decimal: Divide by 100 (move decimal 2 places left).
75% → 75 ÷ 100 = 0.75
Decimal → Fraction: Write over appropriate power of 10, then simplify.
0.75 → 75/100 → ¾
Percentage → Fraction: Write over 100, then simplify.
75% → 75/100 → ¾
Fraction → Percentage: Convert to decimal first, then multiply by 100.
¾ → 0.75 → 75%
Chapter 20: The Three Fundamental Percentage Problems
Every percentage problem you'll ever encounter falls into ONE of three categories.
Type 1 — Finding a Percentage OF a Number
"What is P% of N?"
Formula: Result = (P / 100) × N
Example: What is 15% of 2,400?
Result = (15 / 100) × 2,400
= 0.15 × 2,400
= 360
Real-world use: Sales tax, tips, discounts, interest calculations.
Type 2 — Finding What Percentage One Number Is OF Another
"A is what percent of B?"
Formula: Percentage = (A / B) × 100
Example: You scored 42 out of 60 on a test. What's your percentage?
Percentage = (42 / 60) × 100
= 0.7 × 100
= 70%
Real-world use: Test scores, performance metrics, proportions.
Type 3 — Finding the Original Number When You Know the Percentage
"P% of what number is A?"
Formula: Original = A / (P / 100)
Example: 30% of a number is 450. What's the number?
Original = 450 / (30 / 100)
= 450 / 0.30
= 1,500
Real-world use: Reverse-engineering prices, finding original amounts.
The Three Types at a Glance
| Type | Question | Formula | Example |
|---|---|---|---|
| 1 | What is P% of N? | (P/100) × N | 20% of 500 = 100 |
| 2 | A is what % of B? | (A/B) × 100 | 30 is 60% of 50 |
| 3 | P% of ? = A | A ÷ (P/100) | 25% of 800 = 200 |
Chapter 21: Percentage Increase and Decrease
Percentage Increase
Formula: % Increase = ((New − Original) / Original) × 100
Example: A product's price went from 80 to 92.
Increase = 92 - 80 = 12
% Increase = (12 / 80) × 100 = 15%
To apply an increase:
New = Original × (1 + Percentage/100)
80 × (1 + 15/100) = 80 × 1.15 = 92
Percentage Decrease
Formula: % Decrease = ((Original − New) / Original) × 100
Example: A stock dropped from 250 to 200.
Decrease = 250 - 200 = 50
% Decrease = (50 / 250) × 100 = 20%
To apply a decrease:
New = Original × (1 - Percentage/100)
250 × (1 - 20/100) = 250 × 0.80 = 200
The Asymmetry Trap — A Critical Warning
A percentage increase followed by the same percentage decrease does NOT bring you back to the original number.
Proof:
Start with 100.
Increase by 50%: 100 × 1.50 = 150
Decrease by 50%: 150 × 0.50 = 75
You're at 75, NOT 100.
This is why investors lose more on downturns than they gain on upturns:
| Start | Gain | After Gain | Loss | After Loss | Net Change |
|---|---|---|---|---|---|
| 1,000 | +10% | 1,100 | −10% | 990 | −1% |
| 1,000 | +20% | 1,200 | −20% | 960 | −4% |
| 1,000 | +30% | 1,300 | −30% | 910 | −9% |
| 1,000 | +50% | 1,500 | −50% | 750 | −25% |
| 1,000 | +100% | 2,000 | −100% | 0 | −100% |
Key insight: The bigger the swings, the more damage is done. In finance this is called volatility drag. Remember it.
Chapter 22: Successive Percentages
When percentages are applied one after another:
Multiply the multipliers.
Example: A shirt is marked up 40% then discounted 25%. What's the net change?
Multiplier for 40% increase: 1.40
Multiplier for 25% decrease: 0.75
Combined: 1.40 × 0.75 = 1.05
Net effect: 5% increase
General Formula for Successive Changes:
Final = Original × (1 ± p₁/100) × (1 ± p₂/100) × (1 ± p₃/100) × ...
Successive Percentage Quick Reference
| First Change | Second Change | Combined Multiplier | Net Effect |
|---|---|---|---|
| +10% | +10% | 1.1 × 1.1 = 1.21 | +21% |
| +20% | +20% | 1.2 × 1.2 = 1.44 | +44% |
| +10% | −10% | 1.1 × 0.9 = 0.99 | −1% |
| +20% | −20% | 1.2 × 0.8 = 0.96 | −4% |
| +50% | −30% | 1.5 × 0.7 = 1.05 | +5% |
| −10% | −10% | 0.9 × 0.9 = 0.81 | −19% |
| +25% | +25% | 1.25 × 1.25 = 1.5625 | +56.25% |
Chapter 23: Compound Interest — Percentages Over Time
This is arguably the single most important financial concept you'll ever learn.
Formula: A = P × (1 + r/n)^(n × t)
Where:
- A = Final Amount
- P = Principal (starting amount)
- r = Annual interest rate (as a decimal)
- n = Number of compounding periods per year
- t = Number of years
Simple Interest vs. Compound Interest
Starting with 10,000 at 8% per year:
| Year | Simple Interest | Compound Interest | Difference |
|---|---|---|---|
| 0 | 10,000 | 10,000 | 0 |
| 1 | 10,800 | 10,800 | 0 |
| 2 | 11,600 | 11,664 | 64 |
| 5 | 14,000 | 14,693 | 693 |
| 10 | 18,000 | 21,589 | 3,589 |
| 15 | 22,000 | 31,722 | 9,722 |
| 20 | 26,000 | 46,610 | 20,610 |
| 25 | 30,000 | 68,485 | 38,485 |
| 30 | 34,000 | 100,627 | 66,627 |
After 30 years, compound interest gives you nearly 3× what simple interest gives. This is why Einstein (allegedly) called compound interest "the eighth wonder of the world."
The Rule of 72
To estimate how long it takes your money to double:
Years to Double ≈ 72 / Interest Rate
| Interest Rate | Years to Double |
|---|---|
| 1% | 72 years |
| 2% | 36 years |
| 4% | 18 years |
| 6% | 12 years |
| 8% | 9 years |
| 10% | 7.2 years |
| 12% | 6 years |
| 15% | 4.8 years |
| 20% | 3.6 years |
Chapter 24: Percentages in Real Life
Application 1 — Discounts and Sales
Scenario: An item costs 450 with a 30% discount.
Discount Amount = 30% of 450 = 0.30 × 450 = 135
Sale Price = 450 - 135 = 315
Shortcut: Sale Price = 450 × (1 − 0.30) = 450 × 0.70 = 315
Stacking discounts — a 20% coupon on top of a 30% sale:
After 30% off: 450 × 0.70 = 315
After additional 20% off: 315 × 0.80 = 252
⚠️ This is NOT a 50% discount. It's: 1 − (0.70 × 0.80) = 1 − 0.56 = 44% off.
Application 2 — Tax Calculations
Scenario: Item costs 200, tax is 15%.
Tax = 200 × 0.15 = 30
Total = 200 + 30 = 230
Reverse calculation — finding the pre-tax price.
If the total (with 15% tax included) is 230:
Pre-tax = 230 / 1.15 = 200
General formula: Pre-tax Price = Total / (1 + Tax Rate/100)
Application 3 — Tips
Quick Mental Math Tips:
| Tip % | Mental Math Shortcut |
|---|---|
| 10% | Move decimal one place left |
| 15% | Find 10%, then add half of that |
| 20% | Find 10%, then double it |
| 25% | Divide by 4 |
Example: Bill is 84.60.
10% = 8.46
15% = 8.46 + 4.23 = 12.69
20% = 8.46 × 2 = 16.92
25% = 84.60 / 4 = 21.15
Application 4 — Profit and Loss
Profit % = ((Selling Price − Cost Price) / Cost Price) × 100
Loss % = ((Cost Price − Selling Price) / Cost Price) × 100
| Cost | Selling | Profit/Loss | % |
|---|---|---|---|
| 500 | 650 | Profit: 150 | +30% |
| 1,200 | 900 | Loss: 300 | −25% |
| 80 | 100 | Profit: 20 | +25% |
| 2,000 | 1,700 | Loss: 300 | −15% |
Application 5 — Mixtures and Concentrations
Scenario: Mix 200 ml of a 30% solution with 300 ml of a 50% solution. What's the resulting concentration?
Solute from first: 200 × 0.30 = 60 ml
Solute from second: 300 × 0.50 = 150 ml
Total solute: 60 + 150 = 210 ml
Total volume: 200 + 300 = 500 ml
Concentration = (210 / 500) × 100 = 42%
Application 6 — Population Growth
Formula: Future = Current × (1 + Growth Rate/100)^Years
Example: A city of 2 million grows at 3% annually. Population after 10 years?
Future = 2,000,000 × (1.03)^10
= 2,000,000 × 1.3439
≈ 2,687,800
Chapter 25: Percentage Points vs. Percent
This is where many adults — including news anchors and politicians — get confused.
- Percentage points: the arithmetic difference between two percentages.
- Percent change: the relative change from one percentage to another.
Example: Interest rates go from 4% to 5%.
| Measure | Calculation | Result |
|---|---|---|
| Change in percentage points | 5% − 4% | 1 percentage point |
| Percent change | ((5 − 4) / 4) × 100 | 25% increase |
Saying "rates increased by 1%" is ambiguous. Does it mean rates went to 5%? Or rates went to 4.04%? Always specify percentage points when talking about the arithmetic difference.
Chapter 26: Advanced Percentage Formulas Reference
| Situation | Formula |
|---|---|
| P% of N | (P/100) × N |
| A is what % of B | (A/B) × 100 |
| P% of what is A | A ÷ (P/100) |
| % Increase | ((New − Old) / Old) × 100 |
| % Decrease | ((Old − New) / Old) × 100 |
| After increase | Original × (1 + P/100) |
| After decrease | Original × (1 − P/100) |
| Successive changes | Original × (1 ± p₁/100) × (1 ± p₂/100) |
| Compound Interest | P × (1 + r/n)^(n×t) |
| Reverse % (find original) | Final ÷ (1 ± P/100) |
| Weighted Average % | (Σ weight × value) / Σ weight |
| % Error | ( |
| Markup from Cost | Cost × (1 + Markup%/100) |
| Margin from Selling | SP × (1 − Margin%/100) = Cost |
PART FIVE — PUTTING IT ALL TOGETHER
Chapter 27: Maya's Restaurant Bill — Revisited
Remember the opening scene? Let's solve it with Maya, who's now a completely different person.
The bill: 847.50, split 4 ways, 18% tip.
Step 1: Calculate the tip
18% of 847.50
= 0.18 × 847.50
= 152.55
Step 2: Total with tip
847.50 + 152.55 = 1,000.05
Step 3: Split 4 ways
1,000.05 ÷ 4 = 250.01 per person
(First person pays 250.02 to cover the extra cent)
Maya looked up from her phone and said, "250 and one cent each. I'll throw in the extra cent."
Her friends stared.
"Since when do you..." Raj started.
Maya just smiled.
Chapter 28: The Grand Conversion Chart
Your ultimate at-a-glance reference. Bookmark this.
| Fraction | Decimal | Percentage | Common Use |
|---|---|---|---|
| 1/100 | 0.01 | 1% | Tiny change, rounding error |
| 1/50 | 0.02 | 2% | Small fee, minor adjustment |
| 1/20 | 0.05 | 5% | Standard sales tax, small tip |
| 1/10 | 0.10 | 10% | Easy tip, basic discount |
| 1/8 | 0.125 | 12.5% | Common in trading |
| 1/6 | 0.1667 | 16.67% | One-sixth share |
| 1/5 | 0.20 | 20% | Good tip, standard discount |
| 1/4 | 0.25 | 25% | Quarter, common discount |
| 1/3 | 0.3333 | 33.33% | One-third share |
| 3/8 | 0.375 | 37.5% | Common in stock markets |
| 2/5 | 0.40 | 40% | Less than half |
| 1/2 | 0.50 | 50% | Half-off sale, 50-50 split |
| 3/5 | 0.60 | 60% | Passing grade (some systems) |
| 5/8 | 0.625 | 62.5% | Just over 60% |
| 2/3 | 0.6667 | 66.67% | Two-thirds majority |
| 3/4 | 0.75 | 75% | Three-quarter mark |
| 4/5 | 0.80 | 80% | Strong performance |
| 5/6 | 0.8333 | 83.33% | Five-sixths |
| 7/8 | 0.875 | 87.5% | Nearly complete |
| 9/10 | 0.90 | 90% | Excellent performance |
| 1 | 1.00 | 100% | The whole thing |
| 1 1/4 | 1.25 | 125% | 25% more than the whole |
| 1 1/2 | 1.50 | 150% | Half again as much |
| 2 | 2.00 | 200% | Doubled |
| 3 | 3.00 | 300% | Tripled |
Chapter 29: Mental Math Shortcuts for Percentages
These tricks will make you faster than any calculator.
The "1% and Scale" Method
Find 1% first, then scale to whatever you need.
Example: Find 37% of 800.
1% of 800 = 8
37% = 37 × 8 = 296
The "10% Building Block" Method
| To find... | Do this... | Example (of 650) |
|---|---|---|
| 10% | Move decimal left | 65 |
| 5% | Half of 10% | 32.5 |
| 1% | Move decimal left twice | 6.5 |
| 15% | 10% + 5% | 65 + 32.5 = 97.5 |
| 20% | 10% × 2 | 65 × 2 = 130 |
| 25% | Divide by 4 | 650 / 4 = 162.5 |
| 30% | 10% × 3 | 65 × 3 = 195 |
| 33% | Divide by 3 | 650 / 3 ≈ 216.67 |
| 40% | 10% × 4 | 65 × 4 = 260 |
| 50% | Divide by 2 | 325 |
| 60% | 50% + 10% | 325 + 65 = 390 |
| 75% | 50% + 25% | 325 + 162.5 = 487.5 |
| 90% | 100% − 10% | 650 − 65 = 585 |
The "Flip" Trick
x% of y = y% of x
This means:
- 8% of 50 = 50% of 8 = 4 (much easier)
- 4% of 75 = 75% of 4 = 3 (instant)
- 16% of 25 = 25% of 16 = 4 (magic)
Chapter 30: Expert Insights — Heuristics the Pros Actually Use
These are the patterns that separate the fluent from the merely correct.
1. Always Convert to Multipliers Before Reasoning
Don't think in "% increase" or "% decrease." Think in multipliers. A 25% discount isn't "minus 25" — it's "multiply by 0.75." A 60% markup is "multiply by 1.60." Once everything is a multiplier, every successive-change problem becomes a single multiplication.
2. Anchor on 10% — Then Build
10% is the most useful fraction in adult life. It's one decimal-point shift away. From 10%, you can construct almost any other percentage in seconds: 5% (half of 10%), 15% (10% + 5%), 20% (double 10%), 30% (triple 10%), and so on. Professionals don't compute percentages from scratch — they triangulate from 10%.
3. Recognise the Three Question Shapes
Every percentage question — every single one — is either "find a part of a whole," "find what part one number is of another," or "find the whole when you know a part and its percentage." When you hear a question, classify it into one of the three before reaching for arithmetic. Half the difficulty disappears at classification time.
4. Volatility Drag Is the Most Underrated Math Idea in Finance
Symmetric percentage moves are not symmetric in outcome. A 50% loss requires a 100% gain to recover. A 90% loss requires a 900% gain. This is why position sizing and risk management matter more than picking winners. Most amateur investors don't internalise this until it's too late.
5. Beware the "Of What?" Question
Every percentage is a percentage of something. Whenever someone quotes one — in a headline, a contract, a pitch deck — silently ask "of what?" The same "30% growth" can mean three different things depending on whether the base is the original, the new value, or the industry average. The base is everything.
6. The Reverse Calculation Trap
To remove a percentage that's already been added, you must divide, not subtract. A bill of 230 that includes 15% tax has a pre-tax price of 230 / 1.15 = 200, not 230 − (230 × 0.15) = 195.50. The latter is the most common math mistake in adult life, full stop.
7. Compound Interest Beats Intuition
Human brains are linear; compound growth is exponential. Use the Rule of 72 as your shortcut: years-to-double = 72 / rate. Anything compounding at 7% doubles in roughly a decade. Anything at 1% takes the better part of a lifetime. This single insight reframes most financial decisions.
8. Fractions Beat Decimals for Mental Estimation
When estimating in your head, convert percentages to friendly fractions first. 33% becomes ⅓, 25% becomes ¼, 67% becomes ⅔, 12.5% becomes ⅛. Then divide. "What's 67% of 360?" becomes "⅔ of 360," which is 240 — instantly.
9. Percentage Points Are Not Percent
A move from 4% to 5% is a 1 percentage-point increase, but a 25% relative increase. Journalists conflate these constantly. In contracts, insurance documents, and policy debates, the distinction can be worth millions. Always specify which one you mean.
10. Round Strategically, Not Lazily
Round to make math easy, then correct at the end. 18% tip on 84.60? Round 84.60 to 85, find 20%, that's 17, subtract 2% (about 1.70), giving ~15.30. Total: ~15.20. Close enough for a tip; faster than any phone calculator.
Chapter 31: Common Mistakes and How to Avoid Them
| Mistake | Why It's Wrong | The Fix |
|---|---|---|
| Adding percentages of different bases | 20% of A + 30% of B ≠ 50% of (A+B) | Always calculate on the actual base |
| Assuming % increase + same % decrease = 0 | +50% then −50% = −25% net | Use the multiplier method |
| Confusing % change with percentage points | "Rates rose 2%" vs "rose 2 percentage points" | Specify "percentage points" |
| Dividing by the new number for % decrease | Should divide by the original | Original is always the base |
| Forgetting to convert % to decimal | Writing 15% as 15 in calculations | Always divide by 100 first |
| Treating fractions as two separate numbers | "3/4 means 3 and 4" | It's a single number: 0.75 |
| Subtracting tax from a tax-inclusive total | Total / (1 + rate) gives the right pre-tax figure | Divide, don't subtract |
| Stacking discounts by adding them | 30% + 20% off ≠ 50% off (it's 44%) | Multiply the multipliers |
Chapter 32: Practice Problems — Test Yourself
Level 1: Foundation
- Convert 7/8 to a decimal and percentage.
- What is 35% of 1,200?
- Express 0.045 as a fraction in simplest form.
- Simplify: 48/72.
Level 2: Intermediate
- A population of 45,000 increases by 12%. What's the new population?
- An item originally costs 380. After two successive discounts of 20% and 15%, what's the final price?
- What single discount equals successive discounts of 30% and 20%?
- If 40% of a number is 260, what is 65% of that number?
Level 3: Advanced
- A merchant marks up goods by 60% then offers a 25% discount. What's the net profit percentage?
- A sum doubles in 5 years at compound interest. In how many years will it become 8 times at the same rate?
- The price of an item increases by 20%, then decreases by 20%, then increases by 10%. What's the net change?
- In an election with two candidates, 15% of voters didn't vote. The winner got 60% of the votes cast and won by 2,100 votes. How many eligible voters were there?
Answer Key
1. 7 ÷ 8 = 0.875 = 87.5%
2. 0.35 × 1,200 = 420
3. 0.045 = 45/1000 = 9/200
4. GCF of 48 and 72 = 24 → 48/72 = 2/3
5. 45,000 × 1.12 = 50,400
6. 380 × 0.80 × 0.85 = 380 × 0.68 = 258.40
7. Combined = 1 − (0.70 × 0.80) = 1 − 0.56 = 0.44 → 44% discount
8. 40% of x = 260 → x = 650 → 65% of 650 = 422.5
9. Effective = 1.60 × 0.75 = 1.20 → 20% profit
10. If it doubles in 5 years, it quadruples in 10 years, and becomes 8× in 15 years (since 2³ = 8).
11. 1.20 × 0.80 × 1.10 = 1.056 → 5.6% net increase
12.
Let total eligible voters = V
Votes cast = 0.85V
Winner got 60% = 0.60 × 0.85V = 0.51V
Loser got 40% = 0.40 × 0.85V = 0.34V
Difference = 0.51V − 0.34V = 0.17V = 2,100
V = 2,100 / 0.17 ≈ 12,353 eligible voters
Frequently Asked Questions
Are fractions, decimals, and percentages really the same thing?
Yes — they are three notations for the same idea: a part of a whole. ¾, 0.75, and 75% are identical numbers; they just use different costumes. Fractions emphasise the ratio between parts. Decimals emphasise place value. Percentages emphasise comparison out of 100. Choose the notation that fits the question.
What's the fastest way to convert a fraction to a percentage?
Divide the numerator by the denominator, then multiply by 100. For example, 3/8 = 0.375 = 37.5%. With practice, you'll memorise the common ones: ⅛ = 12.5%, ¼ = 25%, ⅓ ≈ 33.33%, ½ = 50%, ⅔ ≈ 66.67%, ¾ = 75%.
Why doesn't a 50% increase followed by a 50% decrease bring me back to where I started?
Because each percentage is taken on a different base. The 50% increase grows the original by half of itself. The 50% decrease then takes half of the larger number — which is more than half of the original. This is called volatility drag, and it's why a 50% loss requires a 100% gain to recover.
What's the difference between "percent" and "percentage points"?
If interest rates go from 4% to 5%, that's a 1 percentage-point increase but a 25% relative increase ((5 − 4) / 4 × 100). Always specify "percentage points" when talking about arithmetic differences between two percentages.
How do I remove a tax that's already included in a total?
Divide, don't subtract. If a total of 230 includes 15% tax, the pre-tax price is 230 / 1.15 = 200. The formula is Pre-tax = Total / (1 + Tax Rate/100).
What is the Rule of 72?
A shortcut for compound growth: Years to double ≈ 72 / Annual Rate. At 6%, money doubles in roughly 12 years. At 8%, in about 9. It's an approximation, but accurate enough for mental math.
Why do some fractions become repeating decimals?
A fraction terminates as a decimal only when its denominator (in simplest form) has only 2s and/or 5s as prime factors. Any other prime factor — 3, 7, 11, 13 — produces a repeating pattern. That's why ½ = 0.5 (terminates) but ⅓ = 0.333... (repeats forever).
What's the single most useful percentage skill in adult life?
Finding 10% of any number — instantly, by moving the decimal one place to the left. From 10%, you can build 5%, 15%, 20%, 25%, 50%, and most of the percentages you'll ever need in a restaurant, store, or boardroom.
How do stacked discounts actually work?
Multiply the remaining-percentage multipliers. A 30% sale plus a 20% coupon is 0.70 × 0.80 = 0.56, meaning you pay 56% of the original price — a 44% discount, not 50%. Discounts never simply add together.
Is it worth learning this if I have a calculator on my phone?
Yes — for one reason. Calculators answer questions; understanding lets you ask the right questions. You can't punch numbers into a calculator if you don't know what to punch. Every financial decision, contract negotiation, and budget conversation depends on knowing what to compute, not how to compute it.
Epilogue — Maya's Message to You
Six months after that paralysing restaurant moment, Maya did something she never thought possible. She led a budget presentation at work — complete with fractions, percentages, and projections — and her manager said, "This is the clearest financial breakdown anyone on this team has ever made."
Here's what Maya wants you to know:
You're not bad at math. You just never had someone explain it in a way that made sense to you.
The concepts in this guide — numbers, fractions, decimals, and percentages — aren't academic abstractions. They are the language of money, of measurement, of opportunity, of life.
Every time you calculate a tip, every time you understand a loan agreement, every time you evaluate a discount, every time you negotiate a raise — you're using these exact skills.
You now have them.
Your Next Steps
1. Bookmark this post. You'll come back to the conversion tables and formula sheets. Everyone does.
2. Practice one concept today. Just one. Start with the "1% and Scale" method the next time you're at a store.
3. Teach someone else. The best way to cement knowledge is to explain it. Find your own "Maya" and be their Mr. Ade.
If this post helped you see numbers differently, share it with someone who needs to read it. You probably already know who they are.
Quick-Reference Formula Card
╔══════════════════════════════════════════════════════════╗
║ ESSENTIAL FORMULAS ║
╠══════════════════════════════════════════════════════════╣
║ ║
║ Fraction → Decimal: Numerator ÷ Denominator ║
║ Decimal → %: Decimal × 100 ║
║ % → Decimal: Percentage ÷ 100 ║
║ % → Fraction: Percentage / 100, then simplify ║
║ ║
║ P% of N: (P/100) × N ║
║ A is what % of B: (A/B) × 100 ║
║ % Change: ((New − Old) / Old) × 100 ║
║ Compound Interest: P × (1 + r/n)^(n×t) ║
║ Rule of 72: 72 / rate = years to double ║
║ ║
║ x% of y = y% of x (THE FLIP TRICK) ║
║ ║
╚══════════════════════════════════════════════════════════╝
Suggested Internal Links
- The Beginner's Guide to Personal Finance — to extend the compound interest section.
- How to Read a Pay Slip Like a Pro — applies percentage breakdowns to take-home pay.
- The Complete Guide to Budgeting — natural follow-up to Maya's budget example.
- Mental Math for Adults — deeper dive into the shortcuts in Chapters 29–30.
- Understanding Loans, APR, and Interest Rates — pairs with Chapter 23.
Suggested External References
- Khan Academy — Arithmetic & Pre-Algebra (free, comprehensive).
- Better Explained — A Visual Guide to Percentages (intuitive, visual).
- Investopedia — Compound Interest (financial application depth).
- NRICH (University of Cambridge) — Fractions Resources (rigorous teaching content).
- OECD Adult Skills Survey (PIAAC) (data on adult numeracy and why this guide matters).
Written with patience, for anyone who was ever told they "weren't a math person."
You are. You always were.
