3.2 Decimals & Order of Operations
Key Takeaways
- Decimal place values extend base-ten positionally to the right of the decimal point (tenths 10^-1, hundredths 10^-2, thousandths 10^-3), requiring proper place-value alignment during addition and subtraction.
- When multiplying decimals, perform standard integer multiplication and place the decimal point so the product has total decimal places equal to the sum of decimal places in the factors.
- A simplified fraction a/b converts to a terminating decimal if and only if the prime factorization of denominator b contains no prime factors other than 2 and 5; otherwise, it forms a repeating decimal.
- The Order of Operations (PEMDAS/GEMS) establishes strict mathematical hierarchy: Grouping symbols first, Exponents second, Multiplication & Division left-to-right third, and Addition & Subtraction left-to-right fourth.
- Horizontal fraction bars and absolute value bars act as implicit grouping symbols that must be completely evaluated before performing outer operations.
Last updated: July 2026
3.2 Decimals & Order of Operations
Quick Summary: Decimal notation extends the base-ten place value system to fractional parts using powers of ten ($10^{-1}, 10^{-2}, 10^{-3}$). Master exact decimal algorithms, converting between fractions and decimals, identifying terminating vs. repeating decimals, and applying the strict Order of Operations (PEMDAS/GEMS) to complex multi-step numerical expressions.
Decimals and the Order of Operations represent core computational skills tested on Praxis 5003. Elementary teachers must possess complete mastery of positional place value, place-alignment rules, quotient calculations, fraction-decimal equivalences, and grouping hierarchies.
Decimal Place Value & Base-Ten Structure
The decimal system is a positional base-ten system where each place value represents a power of $10$. Positions to the left of the decimal point represent non-negative powers of ten ($10^0 = 1, 10^1 = 10, 10^2 = 100$), while positions to the right represent negative powers of ten (fractional parts):
Reading and Writing Decimals
When reading a decimal, read the whole number part, say "and" for the decimal point, and read the decimal part as a whole number followed by the place value of the last digit.
- Example: $43.085$ is read as "forty-three and eighty-five thousandths."
- Expanded Notation: $43.085 = (4 \times 10^1) + (3 \times 10^0) + (0 \times 10^{-1}) + (8 \times 10^{-2}) + (5 \times 10^{-3})$
Fundamental Decimal Algorithms
1. Addition and Subtraction
Align decimal points vertically so that digits of equal place value are directly above each other. Annex trailing zeros to equalise decimal lengths before computing.
45.200 & \text{(Annex two trailing zeros)} \\ - 18.735 & \text{(Align decimal points vertically)} \\ \hline 26.465 & \end{array}$$ ### 2. Multiplication Multiply numbers as if they were whole integers, ignoring decimal points initially. Count the total number of decimal places across both factors. Place the decimal point in the final product so it has that same total number of decimal places. $$\text{Evaluate: } 3.14 \times 0.25$$ * Factor 1: $3.14$ ($2$ decimal places) * Factor 2: $0.25$ ($2$ decimal places) * Total decimal places required = $2 + 2 = 4$ places. * Integer product: $314 \times 25 = 7850$. * Placing decimal point 4 places from right: $0.7850 = 0.785$. ### 3. Division To divide by a decimal divisor: 1. Shift the decimal point in the divisor to the right until it becomes a whole number. 2. Shift the decimal point in the dividend to the right by the exact same number of places. 3. Place the decimal point in the quotient directly above the new decimal point in the dividend. 4. Perform standard long division. $$\text{Evaluate: } 14.04 \div 0.36$$ * Shift divisor decimal point 2 places right: $0.36 \to 36$. * Shift dividend decimal point 2 places right: $14.04 \to 1404$. * Compute whole number division: $1404 \div 36 = 39$. --- ## Converting Between Fractions & Decimals ### Decimal to Fraction Write the decimal as a fraction with the numerator as the digits and the denominator as the corresponding power of $10$, then simplify to lowest terms. * *Example:* $0.375 = \frac{375}{1000} = \frac{375 \div 125}{1000 \div 125} = \frac{3}{8}$ ### Fraction to Decimal Divide the numerator by the denominator using long division ($a \div b$). ### Terminating vs. Repeating Decimals A rational number in simplest form $\frac{a}{b}$ produces either a **terminating decimal** (ends after a finite number of digits) or a **repeating decimal** (has a repeating block of digits denoted by a vinculum bar $\bar{x}$). > [!IMPORTANT] > **The Prime Factorization Rule:** A fraction $\frac{a}{b}$ in simplest form converts to a **terminating decimal** if and only if the prime factorization of its denominator $b$ contains **no prime factors other than $2$ and $5$** (i.e., $b = 2^m \times 5^n$). If the simplified denominator contains any prime factor other than $2$ or $5$ (such as $3, 7, 11, 13$), the fraction will convert to a **repeating decimal**. | Fraction | Simplest Form | Prime Factorization of Denominator | Decimal Type | Decimal Value | | :--- | :--- | :--- | :--- | :--- | | $\frac{7}{40}$ | $\frac{7}{40}$ | $40 = 2^3 \times 5$ (only 2s & 5s) | Terminating | $0.175$ | | $\frac{9}{15}$ | $\frac{3}{5}$ | $5 = 5^1$ (only 5s) | Terminating | $0.6$ | | $\frac{5}{24}$ | $\frac{5}{24}$ | $24 = 2^3 \times 3$ (contains factor 3) | Repeating | $0.208333... = 0.208\bar{3}$ | | $\frac{9}{14}$ | $\frac{9}{14}$ | $14 = 2 \times 7$ (contains factor 7) | Repeating | $0.64285714...$ | --- ## Order of Operations (PEMDAS / GEMS) When evaluating expressions with multiple operations, mathematicians follow a standardized order of operations to ensure a single, unambiguous result. ### The GEMS Framework While many candidates recall **PEMDAS** (*Please Excuse My Dear Aunt Sally*), modern mathematics instruction uses **GEMS** to avoid common order-of-operation misconceptions: 1. **G – Grouping Symbols:** Evaluate expressions inside grouping symbols from innermost to outermost: Parentheses `()`, Brackets `[]`, Braces `{}`, Absolute Value bars `|x|`, and horizontal Fraction Bars $\frac{\text{Numerator}}{\text{Denominator}}$. 2. **E – Exponents:** Evaluate all exponential terms, powers, and roots. 3. **M – Multiply / Divide:** Perform all multiplication and division strictly **from left to right** as they appear in the expression. 4. **S – Subtract / Add:** Perform all addition and subtraction strictly **from left to right** as they appear in the expression. > [!CAUTION] > **Critical Left-to-Right Equivalence Rule:** Multiplication does NOT take precedence over division, nor does addition take precedence over subtraction! > In the expression $24 \div 4 \times 2$, division comes first from left to right: > $$24 \div 4 \times 2 = 6 \times 2 = 12 \quad (\text{NOT } 24 \div 8 = 3!)$$ --- ## Step-by-Step Worked Exemplar $$\text{Evaluate: } 18 - [3^2 + 2 \times (14 - 4 \times 2)] \div 5$$ 1. **Innermost Grouping Symbol $(14 - 4 \times 2)$:** * Inside parentheses, perform multiplication before subtraction: $4 \times 2 = 8$. * Perform subtraction: $14 - 8 = 6$. * Expression becomes: $18 - [3^2 + 2 \times 6] \div 5$ 2. **Outer Grouping Symbol $[3^2 + 2 \times 6]$:** * Evaluate exponent: $3^2 = 9$. * Evaluate multiplication: $2 \times 6 = 12$. * Evaluate addition: $9 + 12 = 21$. * Expression becomes: $18 - 21 \div 5$ 3. **Division before Subtraction:** * Compute division: $21 \div 5 = 4.2$. 4. **Final Subtraction:** * Compute subtraction: $18 - 4.2 = 13.8$. --- ## Praxis Traps & Common Teacher Misconceptions - **Evaluating PEMDAS Literally:** Executing all multiplications before divisions regardless of position is a frequent error. $12 \div 3 \times 4 = 16$, NOT $1$. - **Ignoring Fraction Bar Groupings:** The fraction bar acts as an implicit grouping symbol. In $\frac{10 + 6}{2 + 2}$, evaluate numerator ($16$) and denominator ($4$) fully before dividing: $16 \div 4 = 4$. - **Misplacing Zeros in Decimal Multiplication:** When multiplying $0.04 \times 0.02$, $4 \times 2 = 8$. With 4 total decimal places, annex leading zeros: $0.0008$.
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Test Your Knowledge
Evaluate the numerical expression using the Order of Operations: $24 \div 4 \times (1 + 2)^2 - | -5 + 2 |$.
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Test Your Knowledge
Which of the following fractions converts to a terminating decimal?
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Test Your Knowledge
A bottle contains $3.75$ liters of fruit juice. If individual serving cups hold $0.125$ liters, how many full servings can be poured from the bottle?
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