Key Takeaways
- Place value: each place is 10 times the place to its right
- Properties: commutative, associative, distributive, identity
- Divide fractions by multiplying by the reciprocal
- PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction
- Multiplication and division are done left to right (same priority level)
Last updated: January 2026
2.1 Numbers and Operations
Numbers and operations form the foundation of elementary mathematics. This section covers about 40% of the Mathematics subtest.
Place Value
Understanding place value is essential:
| Place | Value | Example: 5,432.17 |
|---|---|---|
| Thousands | 1,000 | 5 |
| Hundreds | 100 | 4 |
| Tens | 10 | 3 |
| Ones | 1 | 2 |
| Tenths | 0.1 | 1 |
| Hundredths | 0.01 | 7 |
Key concepts:
- Each place is 10 times the value of the place to its right
- Each place is 1/10 the value of the place to its left
- Zero serves as a placeholder
Operations with Whole Numbers
Addition properties:
- Commutative: a + b = b + a
- Associative: (a + b) + c = a + (b + c)
- Identity: a + 0 = a
Multiplication properties:
- Commutative: a × b = b × a
- Associative: (a × b) × c = a × (b × c)
- Identity: a × 1 = a
- Zero property: a × 0 = 0
- Distributive: a × (b + c) = (a × b) + (a × c)
Fractions
Types of fractions:
- Proper: Numerator < Denominator (3/4)
- Improper: Numerator ≥ Denominator (5/4)
- Mixed number: Whole number + fraction (1 1/4)
Operations with fractions:
| Operation | Same Denominator | Different Denominators |
|---|---|---|
| Add/Subtract | Add/subtract numerators, keep denominator | Find common denominator first |
| Multiply | Multiply numerators, multiply denominators | Same process |
| Divide | Multiply by reciprocal | Same process |
Example: 2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3
Decimals
Converting between fractions and decimals:
- Fraction to decimal: Divide numerator by denominator
- Decimal to fraction: Read the decimal name, simplify
Example:
- 3/4 = 0.75 (3 ÷ 4 = 0.75)
- 0.375 = 375/1000 = 3/8
Order of Operations
PEMDAS/BODMAS:
- Parentheses / Brackets
- Exponents / Orders
- Multiplication and Division (left to right)
- Addition and Subtraction (left to right)
Example: 3 + 4 × 2 = 3 + 8 = 11 (not 14)
Test Your Knowledge
In the number 3,456.78, what is the value of the digit 5?
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Test Your Knowledge
Which property is illustrated by: 3 × (4 + 5) = (3 × 4) + (3 × 5)?
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Test Your Knowledge
What is 2/3 ÷ 4/5?
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Test Your Knowledge
Evaluate: 8 + 2 × 3 - 4
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