5.4 Fraction Multiplication and Division

Learn how to multiply and divide fractions, including steps, examples, and key rules for simplifying results.

Fraction Multiplication and Division covers the procedures for multiplying and dividing fractions, integers, mixed numbers, and signed values together, describing the conceptual meaning of multiplying by a fraction, the mechanical process of combining numerators and denominators, the simplification technique of canceling common factors before multiplying, the reciprocal-based method underlying all fraction division, and the special case introduced by zero.

Fraction Multiplication as a Fraction of a Quantity

Fraction multiplication as a fraction of a quantity interprets multiplying a number by a fraction as taking that fractional part of the number, such as (2/3) × 12 representing two-thirds of 12, which equals 8.

\frac{2}{3} \times 12 = 8

Fraction-by-Fraction Multiplication

Fraction-by-fraction multiplication multiplies the numerators together to form the new numerator and multiplies the denominators together to form the new denominator, such as (2/3) × (3/5) = 6/15, which simplifies to 2/5.

\frac{2}{3} \times \frac{3}{5} = \frac{6}{15} = \frac{2}{5}

Integer and Fraction Multiplication

Integer and fraction multiplication treats an integer as a fraction with a denominator of 1, then applies ordinary fraction-by-fraction multiplication, such as 4 × (3/5) rewritten as (4/1) × (3/5) = 12/5.

4 \times \frac{3}{5} = \frac{12}{5}

Common Factor Cancellation before Multiplication

Common factor cancellation before multiplication reduces a numerator and a denominator appearing anywhere across two fractions being multiplied, before carrying out the multiplication itself, keeping the numbers involved smaller and reducing the need for simplification afterward.

\frac{4}{9} \times \frac{3}{8} = \frac{1}{6}

Signed Fraction Multiplication

Signed fraction multiplication applies the same sign rules established for signed whole-number multiplication—same signs produce a positive product, opposite signs produce a negative product—to the multiplication of two signed fractions.

(- \frac{2}{3}) \times \frac{3}{4} = - \frac{1}{2}

Mixed Number Multiplication

Mixed number multiplication first converts each mixed number into an improper fraction, then applies ordinary fraction-by-fraction multiplication to the resulting improper fractions.

Reciprocal Construction

Reciprocal construction swaps a fraction's numerator and denominator to produce its reciprocal, such as constructing 5/3 as the reciprocal of 3/5, with the product of any nonzero fraction and its reciprocal always equal to 1.

\frac{3}{5} \times \frac{5}{3} = 1

Fraction Division by Reciprocal Multiplication

Fraction division by reciprocal multiplication rewrites a division of fractions as multiplication by the reciprocal of the divisor, such as (2/3) ÷ (4/5) rewritten as (2/3) × (5/4), yielding 10/12, which simplifies to 5/6.

\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{5}{6}

Signed Fraction Division

Signed fraction division rewrites the division as multiplication by the reciprocal of the divisor, then applies the same sign rules established for signed fraction multiplication to the resulting product.

Mixed Number Division

Mixed number division first converts each mixed number into an improper fraction, then applies fraction division by reciprocal multiplication to the resulting improper fractions.

Zero in Fraction Multiplication and Division

Zero in fraction multiplication and division confirms that multiplying any fraction by zero always produces zero, while dividing any fraction by a fraction with a numerator of zero is undefined, since the reciprocal of a fraction with a zero numerator would itself have a zero denominator, which is never permitted.

Fraction Product and Quotient Verification

Fraction product and quotient verification confirms a computed result by converting all values into decimal form and checking consistency, or by applying the appropriate inverse operation—dividing a product back by one of the original factors, or multiplying a quotient back by the divisor—and confirming the original value is recovered.

Together, these procedures establish fraction multiplication as a direct combination of numerators and denominators, simplified where possible through common factor cancellation, and fraction division as multiplication by a reciprocal, both extending consistently to signed values, mixed numbers, and the special restrictions introduced whenever zero is involved.