Fractions, Decimals and Percentages for Engineering Work: THE WORLD OF NUMBERS

Context and scope

Save this post — you'll come back to it.

THE WORLD OF NUMBERS

What Even Is a Number?

Before the practitioner 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 Consider an engineering practitioner's your map of the number universe

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.

The Number Line — Your Secret Weapon

the practitioner's tutor, Mr. the practitioner, 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.)

the practitioner looked at the space between 0 and 1.

"There's... stuff in there?"

"Infinite stuff," Mr. the practitioner smiled. "That's where the magic begins."

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

Example: The number 3,847.56

Total: 3,000 + 800 + 40 + 7 + 0.5 + 0.06 = 3,847.56

Why this matters to you: 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.

FRACTIONS — THE MISUNDERSTOOD HERO

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

Types of Fractions

the practitioner learned there are several kinds:

. Proper Fractions

The numerator is smaller than the denominator.

Always less than 1 whole.

. Improper Fractions

The numerator is equal to or larger than the denominator.

Equal to or greater than 1 whole.

. Mixed Numbers

A whole number combined with a proper fraction.

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 number = 3, Remainder = 2, Denominator stays = 5

Result: 17/5 = 3 ⅖

Equivalent Fractions — The Shape-Shifters

This was the practitioner'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. the practitioner.

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 for Common Values

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 Table

Comparing Fractions

the practitioner 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

Convert both to the same denominator, then compare numerators.

⅜ = 15/40     (multiply top and bottom by 5)
⅖ = 16/40     (multiply top and bottom by 8)

15/40 < 16/40

Therefore: ⅜ < ⅖  ✓

Method 3: Convert to Decimals

⅜ = 3 ÷ 8 = 0.375
⅖ = 2 ÷ 5 = 0.400

0.375 < 0.400

Therefore: 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

Operations with Fractions

This is where the practitioner leveled 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 (Requires Common Denominator):

Step 1: Find the Least Common Denominator (LCD) Step 2: Convert each fraction Step 3: Add the numerators Step 4: Simplify 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

This one is surprisingly the easiest. 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 easier 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 Table

the practitioner Solves the Budget Problem

Remember the practitioner's task? Allocate ⅓ to marketing, ¼ to development, 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):

the practitioner sent the email at 2:47 PM. Thirteen minutes to spare.

She smiled for the first time in years at a math problem.

DECIMALS — FRACTIONS IN DISGUISE

Technical challenge

Two weeks into her journey, the practitioner noticed something: decimals were stalking her.

• Her bank balance: 2,341.87
• The fuel price: 1.459 per liter
• Her fitness app: "You ran 5.3 km today!"
• The weather: "Temperature: 22.5°C"

"Are these... fractions?" she asked Mr. the practitioner.

"Every single one of them," he said.

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

47.8362 = 40 + 7 + 0.8 + 0.03 + 0.006 + 0.0002

Converting Between Fractions and Decimals

Fraction → Decimal

Divide the numerator by the denominator.

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

Quick Reference: Decimals ↔︎ Fractions

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 = 0.142857142857...  written as 0.1̄4̄2̄8̄5̄7̄

The rule: If the denominator (in simplest form) has only factors of 2 and/or 5, the decimal terminates. Otherwise, it repeats.

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 → 7.850

Result: 3.14 × 2.5 = 7.850 = 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

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 UP
• If it's less than 5 → round DOWN (keep it as is)

Example: Round 3.4567 to various places:

PERCENTAGES — THE UNIVERSAL LANGUAGE OF COMPARISON

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:

The Conversion Triangle

This is the single most important skill in practical mathematics.

Fraction → Decimal → Percentage

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 (or move decimal 2 places right).

0.75 → 0.75 × 100 = 75%

Percentage → Decimal:

Divide by 100 (or move decimal 2 places left).

75% → 75 ÷ 100 = 0.75

Decimal → Fraction:

Write as fraction 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%

Complete Conversion Table — Your Permanent Reference

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
Result = 0.15 × 2,400
Result = 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
Percentage = 0.7 × 100
Percentage = 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 Number = A / (P / 100)

Example: 30% of a number is 450. What's the number?

Original = 450 / (30 / 100)
Original = 450 / 0.30
Original = 1,500

Real-world use: Reverse-engineering prices, finding original amounts.

The Three Types at a Glance

Percentage Increase and Decrease

Percentage Increase

Formula: Percentage Increase = ((New Value - Original Value) / Original Value) × 100

Example: A product's price went from 80 to 92.

Increase = 92 - 80 = 12
Percentage Increase = (12 / 80) × 100 = 15%

To apply an increase:

New Value = Original × (1 + Percentage/100)

80 × (1 + 15/100) = 80 × 1.15 = 92

Percentage Decrease

Formula: Percentage Decrease = ((Original Value - New Value) / Original Value) × 100

Example: A stock dropped from 250 to 200.

Decrease = 250 - 200 = 50
Percentage Decrease = (50 / 250) × 100 = 20%

To apply a decrease:

New Value = 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:

Key insight: The bigger the swings, the more damage is done. This is called volatility drag in finance. Remember it.

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

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 times interest is compounded per year
• t = Number of years

Simple Interest vs. Compound Interest Comparison

Starting with 10,000, interest rate of 8% per year:

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

Percentage Applications in Real Life

Application 1: Discounts and Sales

Scenario: An item costs 450, and there's 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 actually: 1 - (0.70 × 0.80) = 1 - 0.56 = 44% off.

Application 2: Tax Calculations

Scenario: Item costs 200, tax rate is 15%.

Tax Amount = 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 Price = 230 / 1.15 = 200

General Formula: Pre-tax Price = Total / (1 + Tax Rate/100)

Application 3: Tips

Quick Mental Math Tips Table:

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

Application 5: Mixtures and Concentrations

Scenario: You mix 200ml of a 30% solution with 300ml of a 50% solution. What's the resulting concentration?

Amount of solute from first:  200 × 0.30 = 60ml
Amount of solute from second: 300 × 0.50 = 150ml
Total solute: 60 + 150 = 210ml
Total volume: 200 + 300 = 500ml

Concentration = (210/500) × 100 = 42%

Application 6: Population Growth and Decline

Formula: Future Population = Current Population × (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
Future = 2,000,000 × 1.3439
Future ≈ 2,687,800

Percentages of Percentages — Percentage Points vs. Percent

This is where many adults — including news anchors and politicians — get confused.

The Difference

• 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%.

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.

Advanced Percentage Formulas Reference

Complete Formula Sheet

PUTTING IT ALL TOGETHER

the practitioner's Restaurant Bill — Revisited

Remember the opening scene? Let's solve it with the practitioner, 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)

the practitioner 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..." the practitioner started.

the practitioner just smiled.