2.1 Operations with Integers, Decimals, Fractions, and Mixed Numbers

Operations with Integers, Decimals, Fractions, and Mixed Numbers

OpenExamPrep provides this quantitative reasoning review to help students master numerical operations tested on the Texas Success Initiative Assessment 2.0 (TSIA2) Mathematics section. On the TSIA2 Mathematics test, foundational arithmetic fluency is evaluated directly. Although an on-screen basic calculator appears on select test items, numerous questions require mental arithmetic and rapid, accurate paper-and-pencil computation. A single sign error or misapplied order-of-operations step can derail an entire multi-step problem.

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1. The Real Number System & Rational Number Classification

Understanding the taxonomy of real numbers provides the vocabulary and rules required for mathematical reasoning on the TSIA2.

Defining Rational Numbers

A rational number is any number that can be expressed as the ratio of two integers:

q = a / b,   where a, b are integers and b ≠ 0

• Every integer k is rational because it can be written as k/1 (e.g., -8 = -8/1).
• Terminating decimals are rational because their place value provides a power-of-10 denominator (e.g., 0.375 = 375/1000 = 3/8).
• Repeating decimals are rational because they can always be converted to fractional form using algebraic techniques (e.g., 0.666... = 2/3; 0.272727... = 27/99 = 3/11).

Absolute Value

The absolute value of a real number x, denoted |x|, represents its geometric distance from zero on the real number line, regardless of direction. Because distance cannot be negative, |x| ≥ 0 for all real numbers:

|x| =  x,   if x ≥ 0
|x| = -x,   if x < 0

When evaluating expressions with absolute value bars, treat the vertical bars as grouping symbols: evaluate the inside arithmetic completely before taking the absolute value.

Example: Evaluate |-14 + 5| - |3 - 11|
Step 1: Simplify inside first bar: |-14 + 5| = |-9| = 9
Step 2: Simplify inside second bar: |3 - 11| = |-8| = 8
Step 3: Subtract: 9 - 8 = 1

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2. Rules of Signs for Integers and Rational Numbers

Sign errors represent the single most common reason students lose points on TSIA2 quantitative items.

Addition and Subtraction

• Same Signs: Add the absolute values (magnitudes) and preserve the common sign.
  • (-7) + (-12) = -19
  • 8 + 14 = 22
• Different Signs: Subtract the smaller absolute value from the larger absolute value; the result carries the sign of the number with the larger absolute value.
  • (-15) + 9 = -(15 - 9) = -6
  • 23 + (-8) = +(23 - 8) = 15
• Subtraction as Adding the Opposite: Every subtraction problem should be conceptually rephrased as adding the additive inverse: a - b = a + (-b).
  • 5 - 18 = 5 + (-18) = -13
  • -9 - (-14) = -9 + 14 = 5
  • -6 - 11 = -6 + (-11) = -17

Multiplication and Division

• The product or quotient of two numbers with identical signs is positive:
  • (+a) × (+b) = +ab
  • (-a) × (-b) = +ab
  • (-36) ÷ (-4) = +9
• The product or quotient of two numbers with opposite signs is negative:
  • (+a) × (-b) = -ab
  • (-a) ÷ (+b) = -(a/b)
  • (-48) ÷ 6 = -8
• Multiple Factors: Count the total number of negative factors:
  • An even number of negative factors produces a positive product: (-2) × (-3) × (-4) × (-5) = +120.
  • An odd number of negative factors produces a negative product: (-2) × (-3) × (-4) = -24.

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3. Order of Operations (GEMDAS / PEMDAS)

Calculations must follow a standardized mathematical hierarchy. On the TSIA2, this hierarchy is often designated GEMDAS (Grouping, Exponents, Multiplication & Division, Addition & Subtraction) to emphasize that grouping symbols extend beyond round parentheses.

⚠️ Critical TSIA2 Rule: Equal Priority Precedence

Multiplication does not take precedence over division, and addition does not take precedence over subtraction! They share equal operational priority. When an expression contains both multiplication and division, you must work strictly from left to right. Similarly, when an expression contains both addition and subtraction, you must compute from left to right.

Step-by-Step Order of Operations Walkthrough

Evaluate the following expression completely:

Problem: -6² - 5|4 - 11| + 36 ÷ 4 × 3

Step 1: Identify grouping symbols.
  The absolute value |4 - 11| acts as a grouping symbol:
  4 - 11 = -7
  |-7| = 7
  Expression becomes: -6² - 5(7) + 36 ÷ 4 × 3

Step 2: Evaluate exponents.
  Notice that -6² means -(6²), not (-6)²:
  6² = 36, so -6² = -36
  Expression becomes: -36 - 5(7) + 36 ÷ 4 × 3

Step 3: Evaluate multiplication and division from left to right.
  First product: 5(7) = 35
  Next, encounter 36 ÷ 4 × 3:
    Evaluate 36 ÷ 4 first = 9
    Then evaluate 9 × 3 = 27
  Expression becomes: -36 - 35 + 27

Step 4: Evaluate addition and subtraction from left to right.
  -36 - 35 = -71
  -71 + 27 = -44

Final Answer: -44

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4. Operations with Fractions and Mixed Numbers

Fraction arithmetic requires systematic manipulation of numerators and denominators.

Converting Between Improper Fractions and Mixed Numbers

• Mixed Number to Improper Fraction: Multiply the integer part by the denominator, add the numerator, and place the result over the original denominator:
  • 4 3/8 = (4 × 8 + 3) / 8 = 35/8
• Improper Fraction to Mixed Number: Perform integer division. The quotient becomes the whole number, the remainder becomes the new numerator, and the denominator remains unchanged:
  • 47 / 6 = 7 R 5 = 7 5/6

Addition and Subtraction: Finding the Least Common Denominator (LCD)

Fractions can only be combined by addition or subtraction when they share a common denominator. The Least Common Denominator (LCD) is the Least Common Multiple (LCM) of the denominators.

Example: Calculate 5/12 + 7/18 - 1/4

Step 1: Find prime factorizations of denominators:
  12 = 2² × 3
  18 = 2 × 3²
  4 = 2²

Step 2: Determine LCD by taking highest power of each prime factor:
  LCD = 2² × 3² = 4 × 9 = 36

Step 3: Convert each fraction to an equivalent fraction with denominator 36:
  5/12 = (5 × 3) / (12 × 3) = 15/36
  7/18 = (7 × 2) / (18 × 2) = 14/36
  1/4  = (1 × 9) / (4 × 9)  = 9/36

Step 4: Combine numerators over the common denominator:
  (15 + 14 - 9) / 36 = 20 / 36

Step 5: Simplify by dividing numerator and denominator by GCD (4):
  20/36 = 5/9

Mixed-Number Subtraction with Regrouping (Borrowing)

When subtracting mixed numbers where the second fraction is larger than the first, you must borrow 1 from the whole number:

Example: Compute 7 1/6 - 3 3/4

Step 1: Find LCD of 6 and 4, which is 12:
  7 1/6 = 7 2/12
  3 3/4 = 3 9/12

Step 2: Regroup because 2/12 is smaller than 9/12:
  Borrow 1 from 7: 7 = 6 + 1 = 6 + 12/12
  Combine: 6 + (12/12 + 2/12) = 6 14/12

Step 3: Subtract whole numbers and fractions separately:
  Whole numbers: 6 - 3 = 3
  Fractions: 14/12 - 9/12 = 5/12
  Result: 3 5/12

Alternative Method (Improper Fractions):
  7 1/6 = 43/6 = 86/12
  3 3/4 = 15/4 = 45/12
  86/12 - 45/12 = 41/12 = 3 5/12

Multiplication and Division of Fractions

• Multiplication: Multiply numerators across and denominators across. Simplify by cross-canceling common factors before multiplying:

  (14 / 25) × (15 / 28)
  Cross-cancel 14 and 28 by dividing both by 14 (leaving 1 and 2).
  Cross-cancel 25 and 15 by dividing both by 5 (leaving 5 and 3).
  = (1 × 3) / (5 × 2) = 3/10
  

• Division (Reciprocal Multiplication): Dividing by a fraction is identical to multiplying by its reciprocal (invert the divisor):

  (a / b) ÷ (c / d) = (a / b) × (d / c)
  
  Example: (9 / 16) ÷ (3 / 8) = (9 / 16) × (8 / 3)
  Cancel common factors:
  9 ÷ 3 = 3; 8 ÷ 16 = 1/2
  = (3 × 1) / (2 × 1) = 3/2 = 1 1/2
  

• Complex Fractions: A fraction whose numerator, denominator, or both contain fractions. Treat the main fraction bar as a division sign:

  [ (1/2) + (2/3) ] / [ (3/4) - (1/6) ]
  Numerator: 1/2 + 2/3 = 3/6 + 4/6 = 7/6
  Denominator: 3/4 - 1/6 = 9/12 - 2/12 = 7/12
  Divide: (7/6) ÷ (7/12) = (7/6) × (12/7) = 12/6 = 2
  

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5. Decimals and Conversions

Decimals represent fractions with base-10 denominators (tenths 10⁻¹, hundredths 10⁻², thousandths 10⁻³, ten-thousandths 10⁻⁴).

Terminating vs. Repeating Decimals

When a simplified fraction a/b is converted to a decimal by long division (dividing a by b):

• Terminating Decimal: The division terminates with a remainder of zero.
  • The Prime Factor Rule: A fully reduced fraction a/b produces a terminating decimal if and only if the prime factorization of its denominator b contains only 2s, only 5s, or both 2s and 5s (i.e., b = 2ᵐ × 5ⁿ for non-negative integers m, n).
  • 3/20: 20 = 2² × 5 → terminates (0.15).
  • 7/40: 40 = 2³ × 5 → terminates (0.175).
• Repeating Decimal: If the denominator of a simplified fraction contains any prime factor other than 2 or 5 (such as 3, 7, 11, 13), the decimal expansion will repeat indefinitely.
  • 5/12: 12 = 2² × 3 → prime factor 3 causes repetition (0.41666... = 0.416̄).
  • 4/15: 15 = 3 × 5 → prime factor 3 causes repetition (0.2666... = 0.26̄).

Essential Fraction-to-Decimal Benchmark Table

Memorizing benchmark equivalents eliminates time-consuming long division on non-calculator items:

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6. TSIA2 Arithmetic Traps & Non-Calculator Strategies

Keep these common pitfalls in mind when solving problems:

1. The Unary Negation Trap (-x² vs (-x)²): On the TSIA2, -5² evaluates to -25 because exponentiation takes precedence over unary negation (-(5²)). Conversely, (-5)² equals +25 because parentheses group the negative sign with the base. When substituting negative values into algebraic expressions like x², always wrap the negative number in parentheses.
2. The Horizontal Fraction Bar as a Grouping Symbol: A fraction bar groups everything in the numerator and everything in the denominator. You must compute (14 + 10) / (2 × 3) as 24 / 6 = 4. Never cancel terms before simplifying the grouped sum in the numerator.
3. Equal Precedence Left-to-Right Violation: In the expression 24 ÷ 6 × 2, calculating 6 × 2 = 12 first yields 24 ÷ 12 = 2 (incorrect). Evaluating strictly left to right gives 24 ÷ 6 = 4, then 4 × 2 = 8 (correct).
4. Fraction Division Inversion Error: In complex fractions, remember to invert the denominator (divisor), not the numerator. In (3/4) / 5, write 5 as 5/1, producing (3/4) × (1/5) = 3/20, not (4/3) × 5 = 20/3.