10.2 Operations with Rational Numbers: Fractions, Decimals, Percents, and Mixed Numbers

10.2 Operations with Rational Numbers: Fractions, Decimals, Percents, and Mixed Numbers

Competency 1.2 of the FTCE Mathematics (828) subtest evaluates a candidate's computational fluency with rational numbers across their primary representations: fractions, decimals, mixed numbers, and percentages. Real-world pedagogical tasks—such as scaling instructional resources, grading weighted student portfolios, and interpreting school performance percentiles—require rapid, accurate arithmetic. Because the test provides only an on-screen four-function calculator, fluent manual work with fractions and conversions is critical to test success.

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Operations with Fractions and Mixed Numbers

Fractions represent parts of a whole or ratios between integers. Performing arithmetic operations on fractions requires distinct protocols depending on whether the operation is additive or multiplicative.

1. Addition and Subtraction: The Common Denominator Requirement

Fractions cannot be added or subtracted directly unless they share identical unit partitions—meaning they must possess a common denominator. The standard procedure employs the Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators.

(a/c) ± (b/c) = (a ± b)/c

When denominators differ, find the LCD, scale the numerators by the respective multiplying factors, and combine:

For 5/6 - 3/8:

• The LCM of 6 and 8 is 24.
• Scale each fraction: (5 × 4)/(6 × 4) = 20/24 and (3 × 3)/(8 × 3) = 9/24.
• Compute the difference: (20 - 9)/24 = 11/24.

The Cross-Multiplication Formula for Two Fractions

For quick scratchpad computation without factoring, you can use the cross-multiplication identity: (a/b) ± (c/d) = (ad ± bc) / (bd) 5/6 - 3/8 = ((5 × 8) - (6 × 3)) / (6 × 8) = (40 - 18) / 48 = 22/48 = 11/24.

2. Mixed Numbers in Addition and Subtraction

Mixed numbers can be combined by grouping whole parts and fractional parts, or by converting both terms to improper fractions. To convert a mixed number W N/D to an improper fraction:

W N/D = ((W × D) + N) / D

Converting to improper fractions is particularly valuable in subtraction to avoid complex fractional regrouping (borrowing):

7 1/5 - 3 3/4 = 36/5 - 15/4 = ((36 × 4) - (15 × 5)) / 20 = (144 - 75) / 20 = 69/20 = 3 9/20.

3. Multiplication: Direct Products and Cross-Cancellation

Multiplication does not require a common denominator. The product of two fractions is simply the product of their numerators divided by the product of their denominators:

(a/b) × (c/d) = (ac) / (bd)

To prevent unwieldy calculations, always cross-cancel common factors between any numerator and any denominator prior to multiplying:

(14/25) × (15/21):

• Divide 14 and 21 by their common factor of 7 (14 becomes 2, 21 becomes 3).
• Divide 15 and 25 by their common factor of 5 (15 becomes 3, 25 becomes 5).
• The expression becomes (2/5) × (3/3) = (2/5) × 1 = 2/5.

Mandatory Rule for Mixed Numbers: Never multiply whole parts and fractional parts separately. You MUST convert all mixed numbers to improper fractions before executing multiplication: 2 2/3 × 1 1/4 = (8/3) × (5/4) = (2/3) × (5/1) = 10/3 = 3 1/3.

4. Division: Reciprocals and "Keep-Change-Flip"

Dividing by a fraction is algebraically equivalent to multiplying by its multiplicative inverse (reciprocal). The reciprocal of a fraction a/b is b/a (where a, b ≠ 0). The foundational algorithm is commonly termed Keep-Change-Flip:

(a/b) ÷ (c/d) = (a/b) × (d/c) = (ad) / (bc)

Keep the first fraction      ──>  a/b
Change division to multiply  ──>  ×
Flip the second fraction     ──>  d/c

Worked Example

4 2/3 ÷ 1 2/5 = (14/3) ÷ (7/5) = (14/3) × (5/7) = (2/3) × (5/1) = 10/3 = 3 1/3.

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Precision Arithmetic with Decimals

Decimals represent fractions with implicit base-10 denominators (10, 100, 1000, …). Precision in decimal arithmetic requires strict adherence to positional place-value rules:

1. Addition and Subtraction

Align the decimal points vertically, pad trailing empty place values with zeros, and execute standard column addition or subtraction, carrying the decimal point straight down into the sum or difference:

For 42.8 - 7.365:

• Align as 42.800 minus 07.365.
• Subtracting through borrowing yields 35.435.

2. Multiplication

Multiply the factors as if they were whole integers, ignoring decimal points initially. Once the integer product is determined, count the total number of decimal digits across both factors. Move the decimal point that total number of places from right to left in the product:

For 3.24 × 0.06:

• Integer product: 324 × 6 = 1944.
• Total decimal places: 2 (in 3.24) + 2 (in 0.06) = 4 places.
• Shift four places left: 0.1944.

3. Division

When the divisor is a decimal, shift its decimal point to the right until it becomes a whole integer. Shift the dividend's decimal point to the right by the exact same number of places (adding trailing zeros if necessary). Then carry the decimal point directly upward into the quotient and perform division:

18.72 ÷ 0.8: Shift both points one place right to yield 187.2 ÷ 8 = 23.4.

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The Tri-Partite Conversion Engine

Fluency on the FTCE requires instantaneous conversion between fractions, decimals, and percentages.

                      ┌────────────────────────┐
                      │        FRACTION        │
                      └─────┬────────────▲─────┘
        Divide Numerator by │            │ Place over 100
        Denominator         │            │ and reduce GCF
                            ▼            │
                      ┌───────────┐    ┌─┴─────────┐
                      │  DECIMAL  ├───>│  PERCENT  │
                      └───────────┘    └───────────┘
                          Multiply by 100 (Shift Right 2)
                          Divide by 100 (Shift Left 2)

Conversion Protocols

1. Fraction to Decimal: Divide numerator by denominator (3 ÷ 8 = 0.375).
2. Decimal to Percent: Multiply by 100 by shifting the decimal point two positions to the right and appending the % symbol (0.375 × 100 = 37.5%).
3. Percent to Decimal: Divide by 100 by shifting the decimal point two positions to the left and removing the % symbol (6.25% ÷ 100 = 0.0625).
4. Percent to Fraction: Write the percentage value over 100 and reduce by dividing out the Greatest Common Factor (45% = 45/100 = 9/20).
5. Decimal to Fraction: Identify the terminal place value (0.035 ends in the thousandths place, giving 35/1000 = 7/200).

High-Yield Benchmark Equivalence Table

Memorizing these core numerical equivalents saves valuable test time on Subtest 4:

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Working with the On-Screen Four-Function Calculator

During Subtest 4: Mathematics, candidates have access to an on-screen four-function calculator (addition, subtraction, multiplication, division). The official materials do not describe parentheses, exponent, or fraction keys, so do not plan on them. Practice with a basic four-function calculator and Pearson's CBT tutorial.

The Sequential Execution Trap

Many basic four-function calculators execute operations in entry order, without applying the algebraic order of operations. For example, if you input: 4 + 6 × 3 = a scientific calculator evaluates 6 × 3 = 18, then 18 + 4 = 22, while an immediate-execution calculator evaluates 4 + 6 = 10, then 10 × 3 = 30. Don't rely on the on-screen calculator to sort out precedence; enter one operation at a time in the correct order.

Strategic Rule: Always compute intermediate multiplication, division, and fractional terms on your scratchpad before adding or subtracting on the on-screen tool. When working with fractions, convert them to 3- or 4-place decimals on scratch paper before performing multi-step operations.