6.1 Arithmetic Operations, Order of Operations & Number Properties

6.1 Arithmetic Operations, Order of Operations & Number Properties

The Mathematics Knowledge (MK) subtest of the AFQT measures your ability to apply fundamental mathematical principles, perform numeric operations, and solve algebraic expressions rapidly and accurately without a calculator. Building absolute mastery over fundamental arithmetic, the order of operations, signed number rules, and number properties is the single most critical step toward securing a top percentile score.

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The Order of Operations: PEMDAS

When evaluating mathematical expressions with multiple operations, following a standardized sequence ensures consistent, unambiguous results. The universal order of operations is governed by the acronym PEMDAS (often remembered via the mnemonic "Please Excuse My Dear Aunt Sally").

The PEMDAS Hierarchy

1. P — Parentheses & Grouping Symbols: Evaluate all operations enclosed within parentheses (), brackets [], braces {} textiles, absolute value bars | |, fraction bars, and radical symbols √ from the innermost grouping outward.
2. E — Exponents & Roots: Evaluate all exponential powers ($x^n$) and square or principal roots ($\sqrt{x}$).
3. M & D — Multiplication & Division: Perform all multiplication and division in order from left to right. Multiplication does not take precedence over division; they hold equal operational priority.
4. A & S — Addition & Subtraction: Perform all addition and subtraction in order from left to right. Addition does not take precedence over subtraction; they hold equal operational priority.

⚠️ CRITICAL AFQT EXAM TRAP: A common mistake on the ASVAB is assuming multiplication must always be performed before division simply because 'M' precedes 'D' in PEMDAS. Consider $24 \div 4 \times 3$. Evaluating left to right yields $6 \times 3 = 18$. Performing multiplication first incorrectly yields $24 \div 12 = 2$.

Step-by-Step Evaluation Example

Evaluate the expression: $32 - 4 \times (5 - 2)^2 + 18 \div 3$

1. Parentheses: Solve $(5 - 2) = 3$. Expression becomes: $32 - 4 \times (3)^2 + 18 \div 3$
2. Exponents: Solve $(3)^2 = 9$. Expression becomes: $32 - 4 \times 9 + 18 \div 3$
3. Multiplication & Division (Left to Right):
   • First: $4 \times 9 = 36 \implies 32 - 36 + 18 \div 3$
   • Second: $18 \div 3 = 6 \implies 32 - 36 + 6$
4. Addition & Subtraction (Left to Right):
   • First: $32 - 36 = -4 \implies -4 + 6$
   • Second: $-4 + 6 = 2$

Final Result = 2

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Fundamental Properties of Real Numbers

Algebraic simplification relies on five core mathematical properties. Recognizing these properties allows you to reorder, regroup, and transform complex AFQT expressions into manageable mental arithmetic.

1. Commutative Property

Changing the order of terms does not change the result. Applies only to addition and multiplication.

• Addition: $a + b = b + a \quad (\text{e.g.}, 7 + 12 = 12 + 7 = 19)$
• Multiplication: $a \cdot b = b \cdot a \quad (\text{e.g.}, 4 \cdot 9 = 9 \cdot 4 = 36)$
• Note: Subtraction and division are not commutative ($8 - 3 \neq 3 - 8$ and $12 \div 4 \neq 4 \div 12$).

2. Associative Property

Changing the grouping of terms does not change the result. Applies only to addition and multiplication.

• Addition: $(a + b) + c = a + (b + c) \quad (\text{e.g.}, (3 + 5) + 4 = 3 + (5 + 4) = 12)$
• Multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c) \quad (\text{e.g.}, (2 \cdot 3) \cdot 5 = 2 \cdot (3 \cdot 5) = 30)$

3. Distributive Property

Multiplying a sum or difference by a value is equivalent to multiplying each addend individually and combining the products.

• Symbolic Form: $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$
• Example: $6(3x - 4) = 18x - 24$
• Negative Distribution: $-3(x - 5) = -3x + 15$ (Remember: negative times negative yields positive).

4. Identity Properties

• Additive Identity: Adding zero to any real number leaves the number unchanged ($a + 0 = a$).
• Multiplicative Identity: Multiplying any real number by one leaves the number unchanged ($a \cdot 1 = a$).

5. Inverse Properties

• Additive Inverse: The sum of any number and its opposite is zero ($a + (-a) = 0$).
• Multiplicative Inverse (Reciprocal): The product of any non-zero number and its reciprocal is one ($a \cdot \frac{1}{a} = 1$).

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Signed Number Arithmetic Rules

Operating with positive and negative integers requires strict adherence to sign rules.

Addition of Signed Numbers

• Same Signs: Add their absolute values and keep the shared sign. $\text{Example: } (-7) + (-8) = -15$
• Different Signs: Subtract the smaller absolute value from the larger absolute value, and assign the sign of the number with the greater absolute value. $\text{Example: } (-12) + 19 = +7 \quad \text{and} \quad 14 + (-22) = -8$

Subtraction of Signed Numbers

Convert subtraction into addition by adding the opposite (additive inverse) of the subtrahend: $a - b = a + (-b)$.

• Example 1: $7 - 15 = 7 + (-15) = -8$
• Example 2: $-9 - (-14) = -9 + 14 = +5$

Multiplication and Division of Signed Numbers

• Same Signs: Product or quotient is always positive ($+ \cdot + = +$ and $- \cdot - = +$). $\text{Examples: } (-6) \times (-7) = +42 \quad \text{and} \quad (-36) \div (-4) = +9$
• Different Signs: Product or quotient is always negative ($+ \cdot - = -$ and $- \cdot + = -$). $\text{Examples: } (-8) \times 5 = -40 \quad \text{and} \quad 48 \div (-6) = -8$

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Prime Factorization, GCF & LCM

Prime vs. Composite Numbers

• A prime number is a whole number greater than 1 with exactly two distinct positive factors: 1 and itself (e.g., $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$). Note: 2 is the only even prime number.
• A composite number has more than two factors (e.g., $4, 6, 8, 9, 10, 12$).
• The numbers 0 and 1 are neither prime nor composite.

Prime Factorization

Prime factorization breaks down a composite number into a product of prime factors. For instance, to find the prime factorization of 360: $360 = 36 \times 10 = (4 \times 9) \times (2 \times 5) = (2^2 \times 3^2) \times (2 \times 5) = 2^3 \times 3^2 \times 5$

Greatest Common Factor (GCF)

The GCF is the largest integer that divides two or more integers without a remainder. To find the GCF:

1. Write out the prime factorization of each number.
2. Identify all shared prime factors.
3. Take the lowest power of each shared factor and multiply them.

$\text{Find GCF of } 72 \text{ and } 108:$ $72 = 2^3 \times 3^2 \quad \text{and} \quad 108 = 2^2 \times 3^3$ $\text{Shared prime bases: } 2 \text{ and } 3 \implies \text{GCF} = 2^2 \times 3^2 = 4 \times 9 = 36$

Least Common Multiple (LCM)

The LCM is the smallest positive integer that is divisible by two or more integers. To find the LCM:

1. Write out the prime factorization of each number.
2. Identify all unique prime bases present in any number.
3. Take the highest power of each prime base and multiply them.

$\text{Find LCM of } 24, 36, \text{ and } 40:$ $24 = 2^3 \times 3^1, \quad 36 = 2^2 \times 3^2, \quad 40 = 2^3 \times 5^1$ $\text{Highest powers: } 2^3, 3^2, 5^1 \implies \text{LCM} = 8 \times 9 \times 5 = 360$