Engineering Arithmetic and Sequence of Operations: A Journey Through Addition, Subtraction, Multiplication &...

A Journey Through Addition, Subtraction, Multiplication & Division

How a struggling shopkeeper, a curious child, and a retired engineer discovered that the entire universe runs on just four operations.

What Are Numbers, Really?

Before the practitioner, the practitioner, or Mr. the practitioner could fix their relationship with math, they needed to go back to the very beginning.

Numbers Are Just a Language

Think about it. When you hold up three fingers, the concept of "three-ness" exists whether or not you have a word for it. Numbers are humanity's way of naming quantities so we can communicate, compare, and calculate.

Here's the hierarchy you need to understand:

The Number Family Tree

                    NUMBERS
                       │
        ┌──────────────┼──────────────┐
        │              │              │
   WHOLE NUMBERS   FRACTIONS      DECIMALS
   (0, 1, 2, 3…)  (1/2, 3/4…)   (0.5, 0.75…)
        │              │              │
        └──────────────┼──────────────┘
                       │
              ALL CONNECTED BY
                       │
           ┌───────┬───┴───┬─────────┐
           │       │       │         │
          ADD   SUBTRACT MULTIPLY  DIVIDE
           +       −       ×         ÷

What this means for you: Whether you're calculating a tip at dinner, splitting rent with roommates, adjusting a recipe, or budgeting your monthly expenses — you're using some combination of these four operations on these three types of numbers. Always. Every single time.

The Three Faces of a Number

Every quantity can wear three masks:

The critical insight: These aren't different numbers. They're the same number in different outfits. Knowing how to move between them is one of the most powerful skills in all of mathematics.

Addition — The Art of Combining

What Addition Actually Is

At its core, addition answers one question: "How much do I have when I put these together?"

The formal definition:

$$a + b = c$$

Where a and b are the addends (the parts) and c is the sum (the whole).

But forget the formula for a moment. Here's what's really happening:

Addition = Combining groups into a single, larger group.

You already do this instinctively. When you grab 3 apples and then grab 2 more, you don't need a calculator to know you have 5 apples. Your brain adds automatically.

The challenge comes when the numbers get bigger, involve decimals, or use fractions. But the principle never changes.

Adding Whole Numbers: The Foundation

The Place Value Secret

Every digit in a number has a position, and that position determines its value.

  Thousands  Hundreds  Tens  Ones
      ↓         ↓       ↓     ↓
      3,        4       5     7

3,457 means:

When you add, you combine digits in the same place value column. That's it. That's the whole secret.

Example: 2,847 + 1,365

    2, 8 4 7
  + 1, 3 6 5
  ─────────
    4, 2 1 2

Step by step:

• Ones: 7 + 5 = 12 → write 2, carry 1
• Tens: 4 + 6 + 1 (carried) = 11 → write 1, carry 1
• Hundreds: 8 + 3 + 1 (carried) = 12 → write 2, carry 1
• Thousands: 2 + 1 + 1 (carried) = 4

Result: 4,212

The Power of Carrying (Regrouping)

"Carrying" isn't a magic trick. It's what happens when a column adds up to more than 9. Since each place can only hold one digit (0–9), the extra value moves to the next higher place.

Think of it like this: if you have 14 ones, that's really 1 ten and 4 ones. So the 1 "carries" over to the tens column.

CARRYING VISUALIZED:

  Column adds to 14
       │
       ▼
  1 ten + 4 ones
   │         │
   ▼         ▼
  Goes to    Stays in
  tens       ones
  column     column

Key engineering insight

Back in the practitioner's shop, Mrs. the practitioner showed her the golden rule:

"Line up the decimal points, then add normally."

Example: 4.75 + 12.30 + 0.95

     4.75
   12.30
  + 0.95
  ──────
   18.00

Step by step:

• Hundredths: 5 + 0 + 5 = 10 → write 0, carry 1
• Tenths: 7 + 3 + 9 + 1 (carried) = 20 → write 0, carry 2
• Ones: 4 + 2 + 0 + 2 (carried) = 8
• Tens: 0 + 1 + 0 = 1

Result: 18.00

the practitioner's eyes went wide. It was just whole-number addition with a dot in it.

The rule you must never forget:

Decimal points line up vertically. Always. If a number doesn't have enough decimal places, add zeros.

Example: Adding 5 + 3.27 → Write 5 as 5.00, then add.

Adding Fractions: the practitioner's Puzzle

Remember the practitioner and his farmer's field? Let's solve it together.

The Problem: 3/4 + 1/8 = ?

The Rule: You can only add fractions when they have the same denominator (bottom number). Why? Because the denominator tells you what size the pieces are.

Imagine a pizza. If one person cuts it into 4 slices and another cuts a different pizza into 8 slices, the slices are different sizes. You can't just add 3 slices + 1 slice and get a meaningful answer — unless the slices are the same size.

Step 1: Find a Common Denominator

The denominators are 4 and 8. The smallest number both divide into evenly is 8 (called the Least Common Denominator, or LCD).

Step 2: Convert the Fractions

3/4 = ?/8

Multiply top and bottom by 2:
3 × 2 = 6
4 × 2 = 8

So: 3/4 = 6/8

Step 3: Add the Numerators

$$\frac{6}{8} + \frac{1}{8} = \frac{6 + 1}{8} = \frac{7}{8}$$

Answer: The farmer has planted 7/8 of his field.

the practitioner stared at his paper. For the first time, it made sense. The pieces had to be the same size before you could count them together.

Adding Fractions: The Complete Method

Here's the universal process that works every time:

$$\frac{a}{b} + \frac{c}{d} = \frac{a \times d + c \times b}{b \times d}$$

Example: 2/3 + 3/5

$$\frac{2}{3} + \frac{3}{5} = \frac{(2 \times 5) + (3 \times 3)}{3 \times 5} = \frac{10 + 9}{15} = \frac{19}{15} = 1\frac{4}{15}$$

Adding Mixed Numbers

A mixed number is a whole number combined with a fraction (like 2 ¾).

Method: Add the wholes, then add the fractions.

Example: 3 ½ + 2 ¾

• Whole numbers: 3 + 2 = 5
• Fractions: 1/2 + 3/4 = 2/4 + 3/4 = 5/4 = 1 ¼
• Combine: 5 + 1 ¼ = 6 ¼

Properties of Addition You Can Count On

These aren't just academic rules — they're shortcuts that make mental math faster:

Mental math trick using these properties:

Want to add 47 + 36 in your head?

• Rearrange: (47 + 3) + 33 = 50 + 33 = 83 ✓

You just used the associative property to make the problem easier. That's not cheating — that's mathematical fluency.

Subtraction — The Art of Finding What's Left

What Subtraction Actually Is

Subtraction answers the question: "How much is left when I take some away?"

$$a - b = c$$

Where a is the minuend (what you start with), b is the subtrahend (what you take away), and c is the difference (what remains).

But subtraction is more versatile than "taking away." It actually solves three types of problems:

THREE FACES OF SUBTRACTION:

1. TAKING AWAY     → "I had 10, spent 4, what's left?"
                      10 - 4 = 6

2. FINDING THE GAP → "How much more is 10 than 4?"
                      10 - 4 = 6

3. COMPARING       → "What's the difference between 10 and 4?"
                      10 - 4 = 6

Same operation, three different real-world meanings. This is why subtraction shows up everywhere — from calculating change to measuring gaps to comparing scores.

Subtracting Whole Numbers: Borrowing Demystified

Example: 803 - 567

    803
  - 567
  ─────
    236

Here's where it gets interesting. In the ones column, you need to do 3 - 7. But 3 is smaller than 7. What now?

Borrowing (Regrouping):

You "borrow" 1 ten from the tens column, converting it into 10 ones:

• Ones: Can't do 3 - 7 → Borrow from tens. But tens column is 0!
• Tens: Can't borrow from 0 → Go to hundreds.
• Hundreds: Borrow 1 hundred → 8 becomes 7, tens become 10
• Tens: Now borrow 1 ten from the 10 → tens become 9, ones become 13
• Ones: 13 - 7 = 6
• Tens: 9 - 6 = 3
• Hundreds: 7 - 5 = 2

Result: 236 ✓

BORROWING CHAIN VISUALIZED:

    8  0  3          7  10  3         7   9  13
  - 5  6  7   →   - 5   6  7   →  - 5   6   7
  ─────────        ──────────       ──────────
                                    2   3   6

Key engineering insight

Mrs. the practitioner taught the practitioner a technique called "counting up" — a subtraction method that doesn't feel like subtraction at all.

Problem: 20.00 - 7.35 = ?

Instead of subtracting, count forward from 7.35 to 20.00:

7.35  → 7.40   (added 0.05)
7.40  → 8.00   (added 0.60)
8.00  → 10.00  (added 2.00)
10.00 → 20.00  (added 10.00)
                ──────────
                Total: 12.65

The change is 12.65. ✓

the practitioner started using this method every day. No borrowing, no stress — just counting forward.

Why this works for you: Next time you need to calculate change or find the gap between two numbers, try counting up. It's the same subtraction — but it feels like addition, which most people find easier.

Subtracting Decimals

Same golden rule as adding: line up the decimal points.

Example: 15.00 - 8.73

   15.00
  - 8.73
  ──────
    6.27

• Hundredths: 0 - 3 → borrow → 10 - 3 = 7
• Tenths: 9 (after borrowing) - 7 = 2
• Ones: 14 (after borrowing) - 8 = 6

Result: 6.27

Subtracting Fractions

Same rule as adding: get a common denominator first, then subtract the numerators.

$$\frac{a}{b} - \frac{c}{d} = \frac{a \times d - c \times b}{b \times d}$$

Example: 5/6 - 1/4

$$\frac{5}{6} - \frac{1}{4} = \frac{(5 \times 4) - (1 \times 6)}{6 \times 4} = \frac{20 - 6}{24} = \frac{14}{24} = \frac{7}{12}$$

Simplification tip: Always check if the answer can be reduced. Here, 14 and 24 are both divisible by 2, giving us 7/12.

Subtracting Mixed Numbers (The Tricky Case)

Example: 5 ⅓ - 2 ¾

Step 1: Convert to improper fractions.

• 5 ⅓ = (5 × 3 + 1)/3 = 16/3
• 2 ¾ = (2 × 4 + 3)/4 = 11/4

Step 2: Find common denominator (12).

• 16/3 = 64/12
• 11/4 = 33/12

Step 3: Subtract.

$$\frac{64}{12} - \frac{33}{12} = \frac{31}{12} = 2\frac{7}{12}$$

Key Properties of Subtraction

Critical warning: Unlike addition, order matters in subtraction. You cannot rearrange the numbers freely. 5 - 3 is NOT the same as 3 - 5.

Multiplication — The Art of Scaling

Key engineering insight

Mr. the practitioner had built bridges across three continents. He'd used multiplication thousands of times in engineering calculations. But when his 8-year-old granddaughter Lena asked, "Grandpa, what IS multiplying?" he realized he'd been using it without truly understanding it.

He thought about it all night. And by morning, he had the clearest explanation he'd ever found:

"Multiplication is repeated addition that got tired of repeating."

What Multiplication Actually Is

$$a \times b = c$$

Where a and b are the factors and c is the product.

But what does it MEAN?

3 × 4 means: "three groups of four"

🍎🍎🍎🍎
🍎🍎🍎🍎
🍎🍎🍎🍎

Count them: 12 apples.

Or equivalently: Add 4 three times: 4 + 4 + 4 = 12

Or: Add 3 four times: 3 + 3 + 3 + 3 = 12

Both give you the same answer. This is the commutative property in action.

The Multiplication Table: Your Best Friend

Before anything else — before fractions, before decimals, before algebra — this table is the single most valuable tool in your mathematical toolkit.

×  │  1    2    3    4    5    6    7    8    9   10   11   12
───┼──────────────────────────────────────────────────────────
1  │  1    2    3    4    5    6    7    8    9   10   11   12
2  │  2    4    6    8   10   12   14   16   18   20   22   24
3  │  3    6    9   12   15   18   21   24   27   30   33   36
4  │  4    8   12   16   20   24   28   32   36   40   44   48
5  │  5   10   15   20   25   30   35   40   45   50   55   60
6  │  6   12   18   24   30   36   42   48   54   60   66   72
7  │  7   14   21   28   35   42   49   56   63   70   77   84
8  │  8   16   24   32   40   48   56   64   72   80   88   96
9  │  9   18   27   36   45   54   63   72   81   90   99  108
10 │ 10   20   30   40   50   60   70   80   90  100  110  120
11 │ 11   22   33   44   55   66   77   88   99  110  121  132
12 │ 12   24   36   48   60   72   84   96  108  120  132  144

Mr. the practitioner's revelation: He finally understood WHY 6 × 7 = 42. It's because if you arrange 6 rows of 7 objects (or 7 rows of 6), you get 42 objects. Not because someone said so — because the geometry demands it.

Multiplying Multi-Digit Numbers: The Standard Algorithm

Example: 247 × 36

      2 4 7
    ×   3 6
    ───────
    1 4 8 2   ← 247 × 6
  7 4 1 0     ← 247 × 30  (shift one position left)
  ─────────
  8 8 9 2

Breaking it down:

Step 1: Multiply 247 by 6 (the ones digit of 36)

• 7 × 6 = 42 → write 2, carry 4
• 4 × 6 = 24, +4 = 28 → write 8, carry 2
• 2 × 6 = 12, +2 = 14 → write 14
• Partial product: 1,482

Step 2: Multiply 247 by 3 (the tens digit of 36, so it represents 30)

• 7 × 3 = 21 → write 1, carry 2
• 4 × 3 = 12, +2 = 14 → write 4, carry 1
• 2 × 3 = 6, +1 = 7 → write 7
• Partial product: 7,410 (shifted left because we're multiplying by 30, not 3)

Step 3: Add the partial products

• 1,482 + 7,410 = 8,892

Multiplying Decimals

The rule: Multiply as if there are no decimals, then count the total decimal places in both original numbers and place the decimal in the answer.

Example: 3.14 × 2.5

Step 1: Multiply 314 × 25 = 7,850

Step 2: Count decimal places:

• 3.14 has 2 decimal places
• 2.5 has 1 decimal place
• Total: 3 decimal places

Step 3: Place decimal → 7.850 = 7.85

DECIMAL PLACEMENT RULE:

  3.14   ← 2 decimal places
× 2.5    ← 1 decimal place
──────     ──────────────────
  7.850  ← 3 decimal places total
  = 7.85

Multiplying Fractions: Simpler Than You Think

Here's a beautiful surprise: multiplying fractions is EASIER than adding them. No common denominator needed.

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$

Multiply the tops. Multiply the bottoms. Done.

Example: 2/3 × 4/5

$$\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}$$

But what does it MEAN to multiply fractions?

Think of it as "a fraction OF a fraction."

2/3 × 4/5 means "two-thirds OF four-fifths."

VISUALIZING 2/3 × 4/5:

Start with 4/5 of a whole (shaded):
┌──────┬──────┬──────┬──────┬──────┐
│██████│██████│██████│██████│      │
└──────┴──────┴──────┴──────┴──────┘

Now take 2/3 of that shaded part:
┌──────┬──────┬──────┬──────┬──────┐
│▓▓▓▓▓▓│▓▓▓▓▓▓│      │      │      │  ← Top 2 rows (2/3)
│▓▓▓▓▓▓│▓▓▓▓▓▓│██████│██████│      │  ← Middle row
│      │      │      │      │      │  ← Bottom row
└──────┴──────┴──────┴──────┴──────┘

The double-shaded (▓) area = 8 out of 15 boxes = 8/15

Multiplying Mixed Numbers

Always convert to improper fractions first.

Example: 2 ½ × 1 ⅓

• 2 ½ = 5/2
• 1 ⅓ = 4/3

$$\frac{5}{2} \times \frac{4}{3} = \frac{20}{6} = \frac{10}{3} = 3\frac{1}{3}$$

Cross-Cancellation: The Speed Trick

Before multiplying fractions, see if any numerator shares a common factor with any denominator. Cancel them first for smaller numbers.

Example: 8/15 × 5/12

• The 8 on top and 12 on bottom share factor 4: 8÷4 = 2, 12÷4 = 3
• The 5 on top and 15 on bottom share factor 5: 5÷5 = 1, 15÷5 = 3

$$\frac{8}{15} \times \frac{5}{12} = \frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$$

Much simpler than multiplying 8 × 5 = 40 and 15 × 12 = 180, then reducing 40/180.

Properties of Multiplication

The Distributive Property is the most powerful tool in this list. It's the bridge between addition and multiplication, and it's how you'll eventually understand algebra.

Mental math example using distributive property:

$$23 \times 7 = (20 \times 7) + (3 \times 7) = 140 + 21 = 161$$

Division — The Art of Sharing and Grouping

Improvement method and result

Six months into running her shop, the practitioner was a different person.

She could add prices in her head, make change without flinching, and calculate bulk discounts using multiplication. But then a customer asked a question that opened the final door:

"If I buy 8 of these and the total is 52, how much is each one?"

the practitioner needed division.

What Division Actually Is

Division answers two questions:

Question 1 — Sharing: "If I split this equally among groups, how much does each group get?"

Question 2 — Grouping: "How many groups of this size can I make?"

$$a \div b = c$$

Where a is the dividend (what you're dividing up), b is the divisor (what you're dividing by), and c is the quotient (the answer).

Division is the INVERSE of multiplication. If 6 × 7 = 42, then 42 ÷ 7 = 6 and 42 ÷ 6 = 7.

MULTIPLICATION AND DIVISION: TWO SIDES OF ONE COIN

    6 × 7 = 42
        ↕
   42 ÷ 7 = 6
   42 ÷ 6 = 7

If you know one, you know all three.

Long Division: Step by Step

Example: 8,472 ÷ 6

       1, 4 1 2
      ┌────────
  6   │ 8, 4 7 2
      - 6
      ───
        2 4
      - 2 4
      ─────
          0 7
        -   6
        ─────
            1 2
          - 1 2
          ─────
              0

The process (Divide, Multiply, Subtract, Bring down — repeat):

1. 6 into 8: Goes 1 time. 1 × 6 = 6. Subtract: 8 - 6 = 2.
2. Bring down 4: Makes 24. 6 into 24 goes 4 times. 4 × 6 = 24. Subtract: 0.
3. Bring down 7: Makes 07. 6 into 7 goes 1 time. 1 × 6 = 6. Subtract: 1.
4. Bring down 2: Makes 12. 6 into 12 goes 2 times. 2 × 6 = 12. Subtract: 0.

Result: 1,412 ✓

Verification: 1,412 × 6 = 8,472 ✓

Division with Remainders

Example: 47 ÷ 5

        9  R 2
      ┌─────
  5   │  47
      - 45
      ────
         2  ← remainder

5 goes into 47 nine times (9 × 5 = 45) with 2 left over.

Three ways to express the answer:

Dividing Decimals

Case 1: Dividing a decimal by a whole number

Just divide normally and place the decimal point directly above.

Example: 14.7 ÷ 3

       4.9
      ┌─────
  3   │ 14.7
      - 12
      ─────
        2 7
      - 2 7
      ─────
          0

Result: 4.9

Case 2: Dividing by a decimal

The trick: Move the decimal point in the divisor to make it a whole number, and move the decimal in the dividend the same number of places.

Example: 7.2 ÷ 0.4

Move decimal 1 place right in both numbers:

0.4 → 4
7.2 → 72

Now divide: 72 ÷ 4 = 18

Why this works: You're multiplying both numbers by the same power of 10 (here, by 10). This doesn't change the quotient, just like 10/2 and 100/20 both equal 5.

$$\frac{7.2}{0.4} = \frac{7.2 \times 10}{0.4 \times 10} = \frac{72}{4} = 18$$

Dividing Fractions: The "Flip and Multiply" Rule

This is one of the most famous rules in math, but few people understand WHY it works.

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$

"Keep the first fraction, flip the second, and multiply."

Example: 3/4 ÷ 2/5

$$\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}$$

But WHY does flipping work?

Division asks: "How many 2/5's fit inside 3/4?"

Dividing by a fraction is the same as multiplying by its reciprocal (the flipped version). Here's the deeper reason:

$$\frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c}$$

When you divide by something, you're really asking "what do I multiply by to get from the divisor to the dividend?" The reciprocal is the multiplier that "undoes" a fraction.

Think of it this way: dividing by 2 is the same as multiplying by ½. Dividing by ¾ is the same as multiplying by 4/3. The reciprocal reverses the operation.

Dividing Mixed Numbers

Convert to improper fractions first, then use "flip and multiply."

Example: 3 ½ ÷ 1 ¼

• 3 ½ = 7/2
• 1 ¼ = 5/4

$$\frac{7}{2} \div \frac{5}{4} = \frac{7}{2} \times \frac{4}{5} = \frac{28}{10} = \frac{14}{5} = 2\frac{4}{5}$$

The Golden Rule of Division

You can NEVER divide by zero.

Why? Because division asks "what number times zero gives you something?" And the answer is: no number works. Zero times anything is zero. The question has no answer.

12 ÷ 0 = ???

This asks: 0 × ??? = 12
But 0 × (anything) = 0
So no answer exists.

DIVISION BY ZERO IS UNDEFINED.

This isn't a rule someone made up. It's a logical impossibility — like asking "what's north of the North Pole?"

Converting Between Fractions, Decimals, and Whole Numbers

The Rosetta Stone of Math

Mr. the practitioner called this the "Rosetta Stone" — the key to translating between the three forms of every number.

Fraction → Decimal

Divide the numerator by the denominator.

Decimal → Fraction

Read the decimal as a fraction of the appropriate power of 10, then simplify.

The rule:

• 1 decimal place → denominator is 10
• 2 decimal places → denominator is 100
• 3 decimal places → denominator is 1,000
• And so on...

Converting Repeating Decimals to Fractions

Example: Convert 0.333... to a fraction.

Let x = 0.333...

Multiply both sides by 10: 10x = 3.333...

Subtract the original: 10x - x = 3.333... - 0.333... 9x = 3 x = 3/9 = 1/3

Example: Convert 0.272727... to a fraction.

Let x = 0.272727...

Since two digits repeat, multiply by 100: 100x = 27.272727...

Subtract: 100x - x = 27 99x = 27 x = 27/99 = 3/11

Essential Conversions Reference Chart

Memorize these. They appear everywhere — in cooking, construction, finance, and daily life.

┌────────────┬──────────┬──────────────┐
│  FRACTION  │ DECIMAL  │  PERCENTAGE  │
├────────────┼──────────┼──────────────┤
│    1/2     │   0.5    │     50%      │
│    1/3     │   0.333… │     33.3…%   │
│    2/3     │   0.666… │     66.6…%   │
│    1/4     │   0.25   │     25%      │
│    3/4     │   0.75   │     75%      │
│    1/5     │   0.2    │     20%      │
│    2/5     │   0.4    │     40%      │
│    3/5     │   0.6    │     60%      │
│    4/5     │   0.8    │     80%      │
│    1/6     │   0.166… │     16.6…%   │
│    5/6     │   0.833… │     83.3…%   │
│    1/8     │   0.125  │     12.5%    │
│    3/8     │   0.375  │     37.5%    │
│    5/8     │   0.625  │     62.5%    │
│    7/8     │   0.875  │     87.5%    │
│    1/10    │   0.1    │     10%      │
│    1/100   │   0.01   │      1%      │
└────────────┴──────────┴──────────────┘

Order of Operations — When Everything Happens at Once

Why You Need Rules for the Order

Look at this expression: 6 + 3 × 2

If you add first: (6 + 3) × 2 = 9 × 2 = 18

If you multiply first: 6 + (3 × 2) = 6 + 6 = 12

Two different answers! Which is correct? 12. And the reason is the agreed-upon order of operations.

The Order (From First to Last)

┌─────────────────────────────────────────┐
│         ORDER OF OPERATIONS             │
│                                         │
│  1. PARENTHESES / BRACKETS  ( )  [ ]   │
│  2. EXPONENTS / POWERS      x²  √x     │
│  3. MULTIPLICATION & DIVISION  × ÷     │
│     (left to right)                     │
│  4. ADDITION & SUBTRACTION   + −        │
│     (left to right)                     │
│                                         │
│  Common Mnemonics:                      │
│  • PEMDAS (US) — "Please Excuse My     │
│    Dear Aunt Sally"                     │
│  • BODMAS (UK/AUS) — "Brackets,        │
│    Orders, Division, Multiplication,    │
│    Addition, Subtraction"              │
│  • BEDMAS (Canada)                      │
└─────────────────────────────────────────┘

Critical note: Multiplication and Division have equal priority — you do them left to right, whichever comes first. Same for Addition and Subtraction.

A Complex Example, Step by Step

Solve: 4 + 6 × (3 + 2)² ÷ 10 - 1

Step 1: Parentheses    → 4 + 6 × (5)² ÷ 10 - 1
Step 2: Exponents      → 4 + 6 × 25 ÷ 10 - 1
Step 3: Multiplication → 4 + 150 ÷ 10 - 1
Step 4: Division       → 4 + 15 - 1
Step 5: Addition       → 19 - 1
Step 6: Subtraction    → 18

Answer: 18

Order of Operations with Fractions

Example: (1/2 + 1/3) × 6 ÷ 2

Step 1: Parentheses → (3/6 + 2/6) × 6 ÷ 2
                     = (5/6) × 6 ÷ 2
Step 2: Multiply    → 5 ÷ 2  (since 5/6 × 6 = 5)
Step 3: Divide      → 5/2 = 2.5

Real-World Applications — Where It All Comes Together

the practitioner's Monthly Inventory

the practitioner now tracks her shop's monthly numbers with confidence. Here's a simplified version of her tracking sheet:

Monthly Revenue Calculation:

Each revenue figure uses multiplication: Price × Quantity = Revenue

The total uses addition: 850 + 608 + 387 + 232.50 = 2,077.50

Profit calculation uses subtraction:

Revenue:                 2,077.50
Cost of goods:         - 1,450.00
Operating expenses:    -   320.00
                       ──────────
Net Profit:               307.50

Profit margin uses division:

$$\text{Profit Margin} = \frac{\text{Net Profit}}{\text{Revenue}} = \frac{307.50}{2077.50} \approx 0.148 = 14.8\%$$

the practitioner's Recipe Adventure

the practitioner grew up and became a chef. He uses fractions every single day.

Scaling a recipe from 4 servings to 10 servings:

Scaling factor: 10 ÷ 4 = 2.5

Every line uses multiplication. The scaling factor itself uses division. And combining all the ingredients into one recipe uses addition.

Mr. the practitioner's Building Project

Mr. the practitioner designs a community garden that is 15.75 meters long and 8.5 meters wide.

Area (multiplication): $$15.75 \times 8.5 = 133.875 \text{ square meters}$$

If the garden is divided into 12 equal plots (division): $$133.875 \div 12 = 11.15625 \text{ square meters per plot}$$

Fencing needed — perimeter (addition and multiplication): $$2 \times (15.75 + 8.5) = 2 \times 24.25 = 48.5 \text{ meters}$$

Cost if fencing is 12.50 per meter (multiplication): $$48.5 \times 12.50 = 606.25$$

Budget remaining after fence if budget is 1,000 (subtraction): $$1000 - 606.25 = 393.75$$