2.3 Multiplication & Division of Fractions and Mixed Numbers

2.3 Multiplication & Division of Fractions and Mixed Numbers

Multiplication and division of fractions are among the most straightforward fraction operations because they do not require finding a common denominator. However, they require careful adherence to specific procedural steps—especially when handling mixed numbers and cross-simplification.

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Fraction Multiplication Fundamentals

To multiply two or more fractions, multiply the numerators straight across to form the new numerator, and multiply the denominators straight across to form the new denominator:

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

Conceptual Meaning of Multiplication

Multiplying a fraction by another fraction means taking a fractional part of a fraction. For instance, $\frac{1}{2} \times \frac{3}{4}$ means finding "half of three-fourths," which equals $\frac{3}{8}$.

                  VISUALIZING FRACTION MULTIPLICATION (1/2 x 3/4)
  
  Divide whole into 4 vertical columns, shade 3 (represents 3/4):
  +---+---+---+---+
  |///|///|///|   |
  +---+---+---+---+
  
  Divide horizontally into 2 equal rows, select top row (represents 1/2 of 3/4):
  +---+---+---+---+
  |XXX|XXX|XXX|   |  --> 3 double-shaded grid regions out of 8 total
  +---+---+---+---+  --> Result = 3/8
  |   |   |   |   |
  +---+---+---+---+

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Cross-Simplifying (Cross-Canceling) Before Multiplying

While you can multiply numerators and denominators directly and simplify the resulting fraction at the end, cross-simplifying beforehand is far more efficient. Because multiplication is commutative ($a \cdot c = c \cdot a$), any factor in any numerator can cancel with an identical factor in any denominator before performing multiplication.

Worked Example 1: Cross-Simplifying Multiplication

Evaluate $\frac{18}{35} \times \frac{25}{24}$ and write the answer in lowest terms.

• Step 1: Look for common factors between numerators (18, 25) and denominators (35, 24).
  • 18 and 24 share a common factor of 6: $18 \div 6 = 3$ and $24 \div 6 = 4$.
  • 25 and 35 share a common factor of 5: $25 \div 5 = 5$ and $35 \div 5 = 7$.
• Step 2: Rewrite the expression with reduced values: $\frac{3}{7} \times \frac{5}{4}$
• Step 3: Multiply straight across: $\frac{3 \times 5}{7 \times 4} = \frac{15}{28}$
• Step 4: Check if $\frac{15}{28}$ can be simplified further. Since GCF(15, 28) = 1, it is in simplest form.

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Reciprocals and Multiplicative Inverses

The reciprocal (or multiplicative inverse) of a fraction $\frac{a}{b}$ is created by interchanging its numerator and denominator to get $\frac{b}{a}$ (where $a, b \neq 0$).

Reciprocal Property: The product of any non-zero number and its reciprocal is always 1:

$\frac{a}{b} \times \frac{b}{a} = \frac{a \cdot b}{b \cdot a} = 1$

• The reciprocal of $\frac{4}{9}$ is $\frac{9}{4}$.
• The reciprocal of a whole number $7 = \frac{7}{1}$ is $\frac{1}{7}$.
• The reciprocal of a mixed number must be found after converting it to an improper fraction: for $2 \frac{1}{3} = \frac{7}{3}$, the reciprocal is $\frac{3}{7}$.

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Fraction Division: The Keep-Change-Flip (KCF) Rule

Dividing by a number is mathematically identical to multiplying by its reciprocal. To divide fractions, use the Keep-Change-Flip (KCF) rule:

1. KEEP the first fraction (the dividend) exactly as it is.
2. CHANGE the division sign ($\div$) to a multiplication sign ($\times$).
3. FLIP the second fraction (the divisor) to its reciprocal.

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

Worked Example 2: Fraction Division

Evaluate $\frac{9}{16} \div \frac{15}{24}$.

• Step 1 (KCF): Keep $\frac{9}{16}$, change $\div$ to $\times$, and flip $\frac{15}{24}$ to $\frac{24}{15}$: $\frac{9}{16} \times \frac{24}{15}$
• Step 2 (Cross-Simplify):
  • 9 and 15 share factor 3: $9 \div 3 = 3$, $15 \div 3 = 5$.
  • 24 and 16 share factor 8: $24 \div 8 = 3$, $16 \div 8 = 2$.
• Step 3 (Multiply): $\frac{3}{2} \times \frac{3}{5} = \frac{3 \times 3}{2 \times 5} = \frac{9}{10}$

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Multiplying and Dividing Mixed Numbers

Mandatory Rule: Never attempt to multiply or divide mixed numbers by operating on whole parts and fraction parts separately. You MUST convert all mixed numbers into improper fractions before multiplying or dividing.

Procedure for Mixed Numbers

1. Convert all mixed numbers to improper fractions.
2. For division, apply Keep-Change-Flip.
3. Cross-simplify common factors.
4. Multiply numerators straight across and denominators straight across.
5. Convert improper results back into mixed numbers in lowest terms.

Worked Example 3: Multiplying Mixed Numbers

Calculate $2 \frac{2}{3} \times 1 \frac{7}{8}$.

• Step 1: Convert both mixed numbers to improper fractions: $2 \frac{2}{3} = \frac{(2 \times 3) + 2}{3} = \frac{8}{3}$ $1 \frac{7}{8} = \frac{(1 \times 8) + 7}{8} = \frac{15}{8}$
• Step 2: Set up multiplication and cross-simplify: $\frac{8}{3} \times \frac{15}{8}$
  • Cancel the 8 in the numerator and denominator: $8 \div 8 = 1$.
  • Cancel 15 and 3: $15 \div 3 = 5$ and $3 \div 3 = 1$.
• Step 3: Multiply remaining terms: $\frac{1 \times 5}{1 \times 1} = \frac{5}{1} = 5$

Worked Example 4: Dividing Mixed Numbers

Calculate $4 \frac{1}{2} \div 1 \frac{4}{5}$.

• Step 1: Convert to improper fractions: $4 \frac{1}{2} = \frac{9}{2} \quad \text{and} \quad 1 \frac{4}{5} = \frac{9}{5}$
• Step 2: Apply Keep-Change-Flip: $\frac{9}{2} \div \frac{9}{5} = \frac{9}{2} \times \frac{5}{9}$
• Step 3: Cross-simplify the 9s (dividing both by 9 to get 1): $\frac{1}{2} \times \frac{5}{1} = \frac{5}{2}$
• Step 4: Convert improper fraction $\frac{5}{2}$ to mixed number: $2 \frac{1}{2}$.

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ACCUPLACER Exam Traps & Common Errors

ACCUPLACER Exam Trap 1: Unnecessary Common Denominator Search
Do not waste valuable test time looking for a common denominator when multiplying or dividing fractions! Common denominators are only required for addition and subtraction.

ACCUPLACER Exam Trap 2: Flipping the Wrong Fraction in Division
In Keep-Change-Flip, only flip the second fraction (the divisor). Flipping the first fraction or flipping both fractions leads to incorrect results.

ACCUPLACER Exam Trap 3: Multiplying Whole Numbers Separately in Mixed Numbers
Writing $3 \frac{1}{2} \times 2 \frac{1}{2} = (3 \times 2) + (\frac{1}{2} \times \frac{1}{2}) = 6 \frac{1}{4}$ is INCORRECT. The true value is $\frac{7}{2} \times \frac{5}{2} = \frac{35}{4} = 8 \frac{3}{4}$.