3.2 Fraction Arithmetic: Adding, Subtracting, Multiplying, & Dividing

3.2 Fraction Arithmetic: Adding, Subtracting, Multiplying, & Dividing

Arithmetic with fractions and mixed numbers represents a major scoring domain on the TABE 13&14 Mathematics test. Workplace problems routinely require combining material dimensions, determining recipe ratios, adjusting electrical load capacities, and dividing bulk goods into equal units. This section details the operational rules, regrouping strategies, and algebraic shortcuts required to perform fraction arithmetic with speed and precision.

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1. Addition and Subtraction with Like Denominators

When fractions share an identical denominator, they represent units of the same partition size. To add or subtract fractions with like denominators:

1. Add or subtract the numerators.
2. Keep the common denominator unchanged.
3. Simplify the resulting fraction to lowest terms or convert to a mixed number if necessary.

$$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d} \qquad \text{and} \qquad \frac{a}{d} - \frac{b}{d} = \frac{a - b}{d} \quad (d \neq 0)$$

Worked Example: Like Denominators

Problem: Compute $\frac{7}{12} + \frac{1}{12} - \frac{2}{12}$.

$$\frac{7 + 1 - 2}{12} = \frac{6}{12} = \frac{6 \div 6}{12 \div 6} = \frac{1}{2}$$

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2. Addition and Subtraction with Unlike Denominators & Finding the LCD

Fractions with different denominators cannot be combined directly because their parts are unequal in size. You must first convert them to equivalent fractions with a shared Least Common Denominator (LCD), which is the Least Common Multiple (LCM) of the denominators.

Algorithm for Unlike Denominators:

1. Find the LCD: Determine the LCM of all denominators using prime factorization or list multiples.
2. Scale Each Fraction: Multiply the numerator and denominator of each fraction by the factor needed to produce the LCD.
3. Add or Subtract Numerators: Perform the operation across the new numerators over the LCD.
4. Simplify: Reduce the final fraction to lowest terms.

Worked Example 1: Addition with Unlike Denominators

Problem: Compute $\frac{5}{6} + \frac{3}{8}$.

1. Find the $\text{LCD}(6, 8)$:
   • Multiples of $6$: $6, 12, 18, \mathbf{24}, 30, \dots$
   • Multiples of $8$: $8, 16, \mathbf{24}, 32, \dots$
   • $\text{LCD} = 24$.
2. Convert fractions: $$\frac{5}{6} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24} \qquad \text{and} \qquad \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24}$$
3. Add numerators: $$\frac{20}{24} + \frac{9}{24} = \frac{29}{24} = 1\frac{5}{24}$$

Worked Example 2: Subtraction with Unlike Denominators

Problem: Compute $\frac{7}{10} - \frac{4}{15}$.

1. Find the $\text{LCD}(10, 15) = 30$.
2. Convert fractions: $$\frac{7}{10} = \frac{7 \times 3}{10 \times 3} = \frac{21}{30} \qquad \text{and} \qquad \frac{4}{15} = \frac{4 \times 2}{15 \times 2} = \frac{8}{30}$$
3. Subtract numerators: $$\frac{21}{30} - \frac{8}{30} = \frac{13}{30}$$

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3. Adding and Subtracting Mixed Numbers & Regrouping (Borrowing)

Adding Mixed Numbers

1. Add whole numbers together.
2. Add the fractional parts using a common denominator.
3. If the fractional sum is improper, convert it to a mixed number and add its whole number part to the total.

$$\text{Example: } 4\frac{2}{3} + 3\frac{3}{4} = 4\frac{8}{12} + 3\frac{9}{12} = (4 + 3) + \frac{17}{12} = 7 + 1\frac{5}{12} = 8\frac{5}{12}$$

Subtracting Mixed Numbers with Regrouping (Borrowing)

When subtracting mixed numbers, if the fraction in the top number (minuend) is smaller than the fraction in the bottom number (subtrahend), you must regroup (borrow) $1$ whole from the top whole number and convert it into fractional form $\frac{D}{D}$:

$$W\frac{N}{D} = (W - 1) + \left(1 + \frac{N}{D}\right) = (W - 1) + \left(\frac{D}{D} + \frac{N}{D}\right) = (W - 1)\frac{N + D}{D}$$

Worked Example: Mixed Number Subtraction with Regrouping

Problem: Compute $8\frac{1}{6} - 3\frac{3}{4}$.

1. Find the LCD of $6$ and $4$, which is $12$.
2. Express both numbers with denominator $12$: $$8\frac{1}{6} = 8\frac{2}{12} \qquad \text{and} \qquad 3\frac{3}{4} = 3\frac{9}{12}$$
3. Notice that $\frac{2}{12} < \frac{9}{12}$. Regroup $1$ from $8$: $$8\frac{2}{12} = 7 + \frac{12}{12} + \frac{2}{12} = 7\frac{14}{12}$$
4. Subtract whole numbers and numerators: $$7\frac{14}{12} - 3\frac{9}{12} = (7 - 3) + \left(\frac{14 - 9}{12}\right) = 4\frac{5}{12}$$

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4. Multiplying Fractions & Cross-Canceling

To multiply fractions, multiply numerators across and denominators across:

$$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \quad (b, d \neq 0)$$

The Cross-Canceling Technique

Before multiplying large numbers, divide out common factors shared between any numerator and any denominator. This technique dramatically speeds up calculations and prevents arithmetic errors.

Worked Example: Cross-Canceling

Problem: Multiply $\frac{14}{25} \times \frac{15}{21}$.

1. Identify common factors diagonally:
   • $14$ (top left) and $21$ (bottom right) share factor $7$: $14 \div 7 = 2$, $21 \div 7 = 3$.
   • $15$ (top right) and $25$ (bottom left) share factor $5$: $15 \div 5 = 3$, $25 \div 5 = 5$.
2. Rewrite the expression: $$\frac{14}{25} \times \frac{15}{21} = \frac{2}{5} \times \frac{3}{3}$$
3. Notice that $\frac{3}{3} = 1$, canceling out the $3$s: $$\frac{2}{5} \times \frac{1}{1} = \frac{2}{5}$$

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5. Multiplying Mixed Numbers

Algorithm for Multiplying Mixed Numbers:

1. Convert all mixed numbers to improper fractions ($W\frac{N}{D} = \frac{W \cdot D + N}{D}$).
2. Express any whole numbers as fractions over $1$ ($k = \frac{k}{1}$).
3. Cross-cancel all common factors.
4. Multiply numerators and denominators.
5. Convert the product back to a mixed number in simplest form.

Worked Example: Multiplying Mixed Numbers

Problem: Compute $3\frac{3}{8} \times 2\frac{2}{9}$.

1. Convert to improper fractions: $$3\frac{3}{8} = \frac{(3 \times 8) + 3}{8} = \frac{27}{8} \qquad \text{and} \qquad 2\frac{2}{9} = \frac{(2 \times 9) + 2}{9} = \frac{20}{9}$$
2. Set up the product and cross-cancel: $$\frac{27}{8} \times \frac{20}{9}$$
   • Divide $27$ and $9$ by $9$: $27 \to 3$, $9 \to 1$.
   • Divide $20$ and $8$ by $4$: $20 \to 5$, $8 \to 2$.
3. Multiply remaining factors: $$\frac{3 \times 5}{2 \times 1} = \frac{15}{2} = 7\frac{1}{2}$$

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6. Dividing Fractions: The Reciprocal and "Keep-Change-Flip"

Division answers the question: How many groups of size $\frac{c}{d}$ fit into $\frac{a}{b}$?

The Multiplicative Inverse (Reciprocal)

The reciprocal of a non-zero fraction $\frac{c}{d}$ is $\frac{d}{c}$, obtained by swapping numerator and denominator. The product of any non-zero number and its reciprocal is always $1$:

$$\frac{c}{d} \times \frac{d}{c} = 1$$

The Keep-Change-Flip (KCF) Rule

Dividing by a fraction is mathematically identical to multiplying by its reciprocal:

$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \cdot d}{b \cdot c}$$

1. Keep the first fraction (dividend) exactly as it is.
2. Change the division sign ($\div$) to multiplication ($\times$).
3. Flip the second fraction (divisor) upside down to its reciprocal.

Worked Example: Dividing Mixed Numbers

Problem: Divide $4\frac{2}{3} \div 1\frac{2}{7}$.

1. Convert to improper fractions: $$4\frac{2}{3} = \frac{14}{3} \qquad \text{and} \qquad 1\frac{2}{7} = \frac{9}{7}$$
2. Apply Keep-Change-Flip: $$\frac{14}{3} \div \frac{9}{7} = \frac{14}{3} \times \frac{7}{9}$$
3. Check for cross-cancellation (none exists between $14, 7$ and $3, 9$): $$\frac{14 \times 7}{3 \times 9} = \frac{98}{27}$$
4. Convert to a mixed number: $98 \div 27 = 3$ with remainder $17 \implies 3\frac{17}{27}$.

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7. Multi-Step Applied Workplace Problems

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TABE Exam Traps & Best Practices