> ## Content Index > Fetch the complete content index at: https://blog.eranorth.com/llms.txt > Use this file to discover other available public pages before exploring further. # A Complete Guide to Numbers, Fractions, Decimals & Percentages - URL: https://blog.eranorth.com/fractions-decimals-percentages-explained/ - Published: 2020-04-01T13:00:00.000Z - Updated: 2026-05-23T11:59:58.000Z - Description: The only guide you'll ever need on numbers, fractions, decimals, and percentages. Built from first principles with real-world examples, conversion tables, mental-math shortcuts, and a story that finally makes math click. - Author: K G J - Tags: Mechanical Design > **Save this post — you'll come back to it.** --- ## 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 1. Convert 7/8 to a decimal and percentage. 2. What is 35% of 1,200? 3. Express 0.045 as a fraction in simplest form. 4. Simplify: 48/72. ### Level 2: Intermediate 1. A population of 45,000 increases by 12%. What's the new population? 2. An item originally costs 380\. After two successive discounts of 20% and 15%, what's the final price? 3. What single discount equals successive discounts of 30% and 20%? 4. If 40% of a number is 260, what is 65% of that number? ### Level 3: Advanced 1. A merchant marks up goods by 60% then offers a 25% discount. What's the net profit percentage? 2. A sum doubles in 5 years at compound interest. In how many years will it become 8 times at the same rate? 3. The price of an item increases by 20%, then decreases by 20%, then increases by 10%. What's the net change? 4. 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.*