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WEST-B

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10.2 Fraction and Decimal Operations, Conversions & Order of Operations

Key Takeaways

  • Adding and subtracting fractions requires finding the Least Common Denominator (LCD); mixed numbers often require regrouping (borrowing a whole from the integer part).
  • Multiplying fractions involves cross-simplifying shared factors prior to multiplying numerators and denominators; fraction division requires multiplying by the divisor's reciprocal ('keep-change-flip').
  • Decimal addition and subtraction mandate strict vertical decimal point alignment; decimal multiplication counts total decimal places in factors, while decimal division shifts the divisor decimal point to create an integer divisor.
  • Repeating decimals convert to exact fractions using algebraic elimination (e.g., 10x - x = 9x), proving they are rational numbers.
  • The standard Order of Operations (PEMDAS) evaluates grouping symbols first, then exponents/roots, followed by multiplication and division (strictly left-to-right), and finally addition and subtraction (strictly left-to-right).
Last updated: August 2026

Fraction and Decimal Operations, Conversions & Order of Operations

Quick Answer: On the WEST-B Mathematics subtest (Objective 0013), operational accuracy with rational numbers in fraction and decimal form is vital. Adding and subtracting fractions requires converting to equivalent fractions with the Least Common Denominator (LCD), while mixed number subtraction often requires regrouping ($7\frac{1}{8} = 6\frac{9}{8}$). Multiplying fractions leverages cross-simplification, and division is performed by multiplying by the reciprocal ($\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$). In decimal arithmetic, align decimal points for addition/subtraction, sum decimal places for products, and shift divisors to whole numbers for quotients. The Order of Operations (PEMDAS) strictly prioritizes Grouping Symbols $\rightarrow$ Exponents $\rightarrow$ Multiplication & Division (equal priority, left-to-right) $\rightarrow$ Addition & Subtraction (equal priority, left-to-right).

1. Fraction Arithmetic & Mixed Number Operations

A fraction $\frac{a}{b}$ represents $a$ equal parts of a whole partitioned into $b$ equal pieces.

Types of Fractions

  • Proper Fraction: Numerator is strictly less than denominator ($|a| < |b|$, e.g., $\frac{3}{5}, \frac{7}{12}$). Value is between $-1$ and $1$.
  • Improper Fraction: Numerator is greater than or equal to denominator ($|a| \ge |b|$, e.g., $\frac{11}{4}, \frac{8}{8}$). Value is $\ge 1$ or $\le -1$.
  • Mixed Number: Consists of an integer part and a proper fraction part ($2\frac{3}{4} = 2 + \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$).
+---------------------------------------------------------------------------------------------------+
|                                FRACTION COMPUTATION MASTER MATRIX                                 |
|                                                                                                   |
|   +-------------------+-----------------------------------+-----------------------------------+   |
|   | OPERATION         | ALGORITHM & CORE RULE             | WORKED STEP-BY-STEP EXAMPLE       |
|   +-------------------+-----------------------------------+-----------------------------------+   |
|   | Addition /        | 1. Find LCD = LCM(d1, d2)         | 5/6 + 3/8                         |   |
|   | Subtraction       | 2. Scale numerators               | LCD(6, 8) = 24                    |   |
|   |                   | 3. Add/subtract numerators only   | (5×4)/24 + (3×3)/24               |   |
|   |                   | 4. Keep common denominator        | = 20/24 + 9/24 = 29/24 = 1 5/24   |   |
|   +-------------------+-----------------------------------+-----------------------------------+   |
|   | Multiplication    | 1. Convert mixed numbers to impr. | 3/8 × 16/21                       |   |
|   |                   | 2. Cross-cancel common factors    | (3÷3)/(8÷8) × (16÷8)/(21÷3)       |   |
|   |                   | 3. Multiply numerators & denom.   | = (1/1) × (2/7) = 2/7             |   |
|   +-------------------+-----------------------------------+-----------------------------------+   |
|   | Division          | 1. Convert mixed numbers to impr. | 5/12 ÷ 15/16                      |   |
|   |                   | 2. Multiply by reciprocal (KCF)   | = 5/12 × 16/15                    |   |
|   |                   | 3. Cross-cancel and multiply      | = (5÷5)/(12÷4) × (16÷4)/(15÷5)    |   |
|   |                   |                                   | = (1/3) × (4/3) = 4/9             |   |
|   +-------------------+-----------------------------------+-----------------------------------+   |
+---------------------------------------------------------------------------------------------------+

Mixed Number Subtraction with Regrouping (Borrowing)

When subtracting mixed numbers where the minuend fraction is smaller than the subtrahend fraction, you must borrow $1$ from the whole number portion and convert it to $\frac{d}{d}$ before subtracting.

Worked Example: Evaluate $7\frac{1}{8} - 3\frac{5}{6}$

  1. Find LCD: Denominators are 8 and 6. $\text{LCM}(8, 6) = 24$.
  2. Convert to equivalent fractions: 718=71×38×3=73247\frac{1}{8} = 7\frac{1 \times 3}{8 \times 3} = 7\frac{3}{24} 356=35×46×4=320243\frac{5}{6} = 3\frac{5 \times 4}{6 \times 4} = 3\frac{20}{24}
  3. Regroup (Borrowing): Since $\frac{3}{24} < \frac{20}{24}$, borrow $1$ from the whole number $7$: 7324=(6+1)+324=6+2424+324=627247\frac{3}{24} = (6 + 1) + \frac{3}{24} = 6 + \frac{24}{24} + \frac{3}{24} = 6\frac{27}{24}
  4. Subtract whole numbers and numerators: 62724−32024=(6−3)+(27−2024)=37246\frac{27}{24} - 3\frac{20}{24} = (6 - 3) + \left(\frac{27 - 20}{24}\right) = \mathbf{3\frac{7}{24}}

2. Decimal Arithmetic Mechanics

Addition and Subtraction

  • Align digits vertically strictly by their decimal points.
  • Pad empty trailing place values with zeros to prevent place-value misalignment.
  • Example: Evaluate $42.7 - 5.846$ 
       42.700  (padded with zeros)
 -  5.846
 --------
   36.854

Decimal Multiplication

  1. Multiply the numbers as if they were whole numbers, ignoring decimal points initially.
  2. Count the total number of decimal places in all factors combined.
  3. Place the decimal point in the product so that it contains that total number of decimal places (counting from right to left).
  • Example: Evaluate $3.45 \times 0.06$
    • Multiply whole numbers: $345 \times 6 = 2,070$.
    • Count decimal places: $3.45$ has 2 places; $0.06$ has 2 places $\implies 2 + 2 = 4$ total decimal places.
    • Place decimal point 4 digits from right: $0.2070 = \mathbf{0.207}$.

Decimal Division

  1. If the divisor is a decimal, move its decimal point to the right until it is a whole number.
  2. Move the decimal point in the dividend to the right by the exact same number of places (append zeros if necessary).
  3. Place the decimal point in the quotient directly above the new decimal point in the dividend and perform standard long division.
  • Example: Evaluate $1.92 \div 0.08$
    • Divisor has 2 decimal places: shift right 2 places $\rightarrow 8$.
    • Dividend shifted right 2 places $\rightarrow 192$.
    • $192 \div 8 = \mathbf{24}$.

3. Conversions Between Fractions, Decimals & Percents

+---------------------------------------------------------------------------------------------------+
|                             RATIONAL CONVERSION PATHWAYS                                          |
|                                                                                                   |
|   Fraction  -----[ Divide Numerator by Denominator ]----->  Decimal                               |
|   Decimal   -----[ Multiply by 100 & Add '%' Sign ]------>  Percent                               |
|   Percent   -----[ Remove '%' & Divide by 100 ]---------->  Decimal                               |
|   Decimal   -----[ Place over 10^k & Reduce ]------------>  Fraction                              |
+---------------------------------------------------------------------------------------------------+

High-Yield Benchmark Equivalence Table

FractionDecimalPercentFractionDecimalPercent
$\frac{1}{2}$$0.5$$50%$$\frac{1}{6}$$0.1\overline{6}$$16.\overline{6}%$ or $16\frac{2}{3}%$
$\frac{1}{3}$$0.\overline{3}$$33.\overline{3}%$ or $33\frac{1}{3}%$$\frac{5}{6}$$0.8\overline{3}$$83.\overline{3}%$ or $83\frac{1}{3}%$
$\frac{2}{3}$$0.\overline{6}$$66.\overline{6}%$ or $66\frac{2}{3}%$$\frac{1}{8}$$0.125$$12.5%$ or $12\frac{1}{2}%$
$\frac{1}{4}$$0.25$$25%$$\frac{3}{8}$$0.375$$37.5%$ or $37\frac{1}{2}%$
$\frac{3}{4}$$0.75$$75%$$\frac{5}{8}$$0.625$$62.5%$ or $62\frac{1}{2}%$
$\frac{1}{5}$$0.2$$20%$$\frac{7}{8}$$0.875$$87.5%$ or $87\frac{1}{2}%$
$\frac{2}{5}$$0.4$$40%$$\frac{1}{10}$$0.1$$10%$
$\frac{3}{5}$$0.6$$60%$$\frac{1}{12}$$0.08\overline{3}$$8.\overline{3}%$ or $8\frac{1}{3}%$
$\frac{4}{5}$$0.8$$80%$$\frac{1}{20}$$0.05$$5%$

Converting Repeating Decimals to Fractions (Algebraic Method)

To convert a periodic decimal to an exact simplified fraction:

  1. Set $x$ equal to the repeating decimal.
  2. Multiply $x$ by $10^m$ (where $m$ is the number of non-repeating digits before the period) to isolate the repeating part.
  3. Multiply $x$ by $10^n$ (where $n$ is total digits to shift through one full repeating block).
  4. Subtract the two equations to eliminate the infinitely repeating tail, then solve for $x$ and simplify.

Worked Example: Convert $0.4\overline{16}$ to a Simplified Fraction

  • Let $x = 0.4161616\dots$
  • Multiply by 10 (1 non-repeating digit): $10x = 4.161616\dots$
  • Multiply by 1,000 (3 digits to shift one period): $1,000x = 416.161616\dots$
  • Subtract: 1,000x−10x=416.1616⋯−4.1616…1,000x - 10x = 416.1616\dots - 4.1616\dots 990x=412  ⟹  x=412990990x = 412 \implies x = \frac{412}{990}
  • Divide numerator and denominator by $\text{GCF}(412, 990) = 2$: x=412÷2990÷2=206495x = \frac{412 \div 2}{990 \div 2} = \mathbf{\frac{206}{495}}

4. The Standard Order of Operations (PEMDAS)

The standard algebraic hierarchy ensures a single, unambiguous value for any numerical expression.

+---------------------------------------------------------------------------------------------------+
|                             ORDER OF OPERATIONS HIERARCHY (PEMDAS)                                |
|                                                                                                   |
|   LEVEL 1: Parentheses & Grouping Symbols                                                         |
|   - Innermost to outermost: ( ), [ ], { }, absolute values | |, radical bars √, fraction bars.     |
|                                                                                                   |
|   LEVEL 2: Exponents & Roots                                                                      |
|   - Evaluate powers and roots from left to right: x², √x, 2³, etc.                                |
|                                                                                                   |
|   LEVEL 3: Multiplication & Division (STRICTLY LEFT TO RIGHT)                                     |
|   - Equal priority! Whichever appears first moving left to right is executed first.               |
|   - NOT multiplication before division! 12 ÷ 3 × 2 = 4 × 2 = 8 (NOT 12 ÷ 6 = 2).                  |
|                                                                                                   |
|   LEVEL 4: Addition & Subtraction (STRICTLY LEFT TO RIGHT)                                         |
|   - Equal priority! Whichever appears first moving left to right is executed first.               |
|   - NOT addition before subtraction! 15 - 7 + 4 = 8 + 4 = 12 (NOT 15 - 11 = 4).                   |
+---------------------------------------------------------------------------------------------------+

The Hidden Grouping Bar in Fractions

In rational expressions of the form $\frac{\text{Numerator}}{\text{Denominator}}$, the horizontal fraction bar acts as a grouping symbol grouping the entire numerator and the entire denominator:

a+bc+d≡(a+b)÷(c+d)\frac{a + b}{c + d} \equiv (a + b) \div (c + d)

You must simplify the entire numerator and the entire denominator completely before performing the final division.

Worked Example: Evaluate Multi-Tiered Arithmetic Expression

Evaluate: 42−2×(7−10)2∣−15+3∣÷2+1\text{Evaluate: } \frac{4^2 - 2 \times (7 - 10)^2}{|-15 + 3| \div 2 + 1}
  1. Simplify Numerator:
    • Inner grouping: $(7 - 10) = -3$.
    • Powers: $(-3)^2 = 9$ and $4^2 = 16$.
    • Expression: $16 - 2 \times 9$.
    • Multiplication: $2 \times 9 = 18$.
    • Subtraction: $16 - 18 = \mathbf{-2}$.
  2. Simplify Denominator:
    • Absolute value grouping: $|-15 + 3| = |-12| = 12$.
    • Expression: $12 \div 2 + 1$.
    • Division before addition: $12 \div 2 = 6$.
    • Addition: $6 + 1 = \mathbf{7}$.
  3. Final Division: NumeratorDenominator=−27\frac{\text{Numerator}}{\text{Denominator}} = \mathbf{-\frac{2}{7}}

5. Diagnostic Error Analysis & Educator Strategies

Candidate Error / MisconceptionMathematical Reality & Diagnostic ExplanationCorrect Pedagogical Intervention
Adding fractions across numerators and denominators: $\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$.Adding requires equal partition sizes (common denominator). $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \neq \frac{2}{5}$.Use fraction bar manipulatives or pizza slice visual models to demonstrate that denominators establish unit size.
Performing multiplication before division regardless of order: $24 \div 6 \times 2 = 24 \div 12 = 2$.Multiplication and division share equal priority and must be executed strictly from left to right: $24 \div 6 \times 2 = 4 \times 2 = 8$.Teach the acronym as P-E-[MD]-[AS] to visually bracket MD and AS as co-equal partner operations.
Misplacing the decimal in products: $0.3 \times 0.2 = 0.6$.$0.3$ (1 place) $\times 0.2$ (1 place) $= 0.06$ (2 places). Thinking $0.3 \times 0.2 = \frac{3}{10} \times \frac{2}{10} = \frac{6}{100} = 0.06$.Have candidates convert decimals to fractions to verify the magnitude of the denominator.
Forgetting to borrow in mixed number subtraction: $5\frac{1}{4} - 2\frac{3}{4} = 3\frac{2}{4}$.Candidates incorrectly subtract smaller numerator from larger ($3 - 1 = 2$). Minuend must borrow 1 whole: $5\frac{1}{4} = 4\frac{5}{4} \implies 4\frac{5}{4} - 2\frac{3}{4} = 2\frac{2}{4} = 2\frac{1}{2}$.Require explicit rewrite of regrouped mixed numbers before subtracting.
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Order of Operations (PEMDAS) Sequential Flowchart
Test Your Knowledge

Evaluate the following arithmetic expression: [4² - 2 × (7 - 10)²] / [|-15 + 3| ÷ 2 + 1]

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Which simplified fraction in lowest terms is exactly equal to the repeating decimal 0.4161616... (where only digits 1 and 6 repeat)?

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