Arithmetic Fundamentals

Master the building blocks of all mathematics: the four basic operations and how to work with fractions, decimals, and percentages.

The Four Basic Operations

Addition (+)

Concept: Combining quantities together.

Life Examples:

• Total grocery bill: $15.50 + $23.75 + $8.90 = $48.15
• Total work hours: 8 + 7.5 + 9 + 8 + 6 = 38.5 hours

Tips:

• Round to estimate first: 16 + 24 + 9 ≈ 49 (close to actual $48.15)
• Add from left to right or group convenient numbers (8 + 6 = 14, then add others)

Subtraction (−)

Concept: Finding the difference between quantities.

Life Examples:

• Change from purchase: $50 − $37.25 = $12.75
• Weight loss: 185 lbs − 172 lbs = 13 lbs lost

Tips:

• Use addition to check: $37.25 + $12.75 = $50 ✓
• When subtracting from round numbers, it's easier to add up from the smaller number

Multiplication (×)

Concept: Repeated addition or scaling.

Life Examples:

• Weekly expense: $15/day × 7 days = $105/week
• Area of room: 12 ft × 15 ft = 180 sq ft

Quick Mental Tricks:

• By 10: Add a zero (45 × 10 = 450)
• By 5: Multiply by 10, then divide by 2 (24 × 5 = 240 ÷ 2 = 120)
• By 9: Multiply by 10, then subtract original (17 × 9 = 170 − 17 = 153)

Division (÷)

Concept: Splitting into equal parts or finding how many times one number fits into another.

Life Examples:

• Per person cost: $180 ÷ 6 people = $30/person
• Miles per gallon: 360 miles ÷ 12 gallons = 30 mpg

Tips:

• Use multiplication to check: 30 × 6 = 180 ✓
• Simplify before dividing: 360 ÷ 12 = 36 ÷ 1.2 = 30

Order of Operations (PEMDAS)

Parentheses → Exponents → Multiplication & Division (left to right) → Addition & Subtraction (left to right)

Example: Calculate 5 + 3 × 2²

1. Exponent first: 2² = 4
2. Multiply: 3 × 4 = 12
3. Add: 5 + 12 = 17

Business Application: Calculate profit: Revenue − (Cost per unit × Units sold) $10,000 − ($15 × 200) = $10,000 − $3,000 = $7,000

Fractions

Understanding Fractions

A fraction represents parts of a whole: numerator/denominator

Real-World Example:

• ¾ of pizza remaining = 3 slices out of 4 slices
• ⅝ of project complete = 5 tasks done out of 8 total tasks

Adding and Subtracting Fractions

Rule: Need common denominators

Example: ½ + ⅓

1. Find common denominator (lowest: 6)
2. Convert: 3/6 + 2/6
3. Add numerators: 5/6

Life Example: Recipe adjustment

• Original calls for ⅔ cup flour
• Add ¼ cup more: ⅔ + ¼ = 8/12 + 3/12 = 11/12 cup

Multiplying Fractions

Rule: Multiply across (numerator × numerator, denominator × denominator)

Example: ⅔ × ¾ = (2 × 3)/(3 × 4) = 6/12 = ½

Life Example: Sale discount

• Item costs $60
• On sale for ⅓ off: $60 × ⅓ = $20 off (pay $40)

Dividing Fractions

Rule: Multiply by the reciprocal (flip the second fraction)

Example: ½ ÷ ¼ = ½ × 4/1 = 4/2 = 2

Life Example: Recipe servings

• Recipe makes ½ batch
• Each serving is ¼ of batch
• ½ ÷ ¼ = 2 servings

Decimals

Understanding Decimals

Decimals are another way to represent fractions, based on powers of 10.

Adding and Subtracting Decimals

Rule: Line up decimal points

  23.45
+  8.9
-------
  32.35

Life Example: Total bill $23.45 + $8.90 + $15.00 = $47.35

Multiplying Decimals

Rule: Multiply as whole numbers, then count total decimal places

Example: 2.5 × 3.2

• Multiply: 25 × 32 = 800
• Count decimals: 2.5 (1 place) + 3.2 (1 place) = 2 total places
• Result: 8.00 = 8

Life Example: Tax calculation

• Purchase: $45.50
• Tax rate: 0.08 (8%)
• Tax: $45.50 × 0.08 = $3.64

Dividing Decimals

Rule: Move decimal in divisor to make it whole, move same places in dividend

Example: 12.6 ÷ 0.3

• Move one place: 126 ÷ 3 = 42

Life Example: Unit price

• $12.60 for 3 pounds
• Per pound: $12.60 ÷ 3 = $4.20/lb

Percentages

Understanding Percentages

"Percent" means "per hundred": it's a fraction with denominator 100.

Converting Between Forms

Percentage to Decimal: Divide by 100 (move decimal 2 places left)

• 35% = 0.35
• 8% = 0.08
• 125% = 1.25

Decimal to Percentage: Multiply by 100 (move decimal 2 places right)

• 0.75 = 75%
• 0.03 = 3%
• 1.5 = 150%

Fraction to Percentage: Divide numerator by denominator, multiply by 100

• ¾ = 0.75 = 75%
• ⅝ = 0.625 = 62.5%

Calculating Percentages

Find percentage of a number: Multiply

Example: What is 15% of $80?

• 0.15 × $80 = $12

Find what percentage one number is of another: Divide and multiply by 100

Example: $20 is what percent of $80?

• ($20 ÷ $80) × 100 = 0.25 × 100 = 25%

Mental Math Shortcuts for Percentages

10% Method (foundation for quick calculations):

• 10% of any number: Move decimal one place left
• 10% of $450 = $45

Building from 10%:

• 20% = 10% × 2
• 5% = 10% ÷ 2
• 15% = 10% + 5%
• 30% = 10% × 3

Example: 15% tip on $64 bill

1. 10% of $64 = $6.40
2. 5% of $64 = $3.20 (half of 10%)
3. 15% = $6.40 + $3.20 = $9.60

1% Method (for small percentages):

• 1% of any number: Move decimal two places left
• 1% of $3,500 = $35
• 3% of $3,500 = $35 × 3 = $105

Practice Problems

Basic Operations

1. $45.75 + $23.50 + $12.00 = ?
2. $100 − $67.43 = ?
3. 25 × $4.50 = ?
4. $156 ÷ 12 = ?

Fractions

5. ½ + ⅓ = ?
6. ⅔ × $90 = ?
7. ¾ ÷ ¼ = ?

Decimals

8. 15.5 × 2.4 = ?
9. $18.75 ÷ 2.5 = ?

Percentages

10. What is 25% of $120?
11. $15 is what percent of $60?
12. Calculate 18% tip on $45 bill

Solutions

1. $81.25
2. $32.57
3. $112.50
4. $13
5. 5/6
6. $60
7. 3
8. 37.2
9. $7.50
10. $30
11. 25%
12. $8.10 (10% = $4.50, 8% = $3.60, total = $8.10)

Key Takeaways

✓ Master the basics first: all advanced math builds on arithmetic
✓ Always estimate: a quick mental check prevents big errors
✓ Understand percentages: they appear in every financial decision
✓ Practice mental math: it saves time and builds confidence
✓ Check your work: use inverse operations (add to check subtraction, etc.)

Real-World Applications

• Budgeting: Adding expenses, calculating totals
• Shopping: Comparing unit prices, calculating discounts
• Cooking: Scaling recipes, converting measurements
• Travel: Converting currencies, calculating distances
• Work: Time tracking, expense reports, billing

Next Steps

Once comfortable with these fundamentals, move to Chapter 02: Algebra Basics to learn how to work with variables and solve equations. This is essential for financial planning and business analysis.