Summary of Fraction Operations
Summary of Fraction Operations consolidates all four operations — multiplication, division, addition, and subtraction — into a single reference for pre-algebra students. Multiplication: multiply numerators and denominators directly, no LCD needed. Division: multiply by the reciprocal. Addition and subtraction: convert to equivalent fractions with the LCD, then operate on numerators. From OpenStax Prealgebra 2E, this summary helps students choose the right procedure quickly, avoiding the common mistake of finding an LCD when multiplying fractions.
Key Concepts
Property Fraction multiplication : Multiply the numerators and multiply the denominators. $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$ Fraction division : Multiply the first fraction by the reciprocal of the second. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$$ Fraction addition/subtraction : Convert to equivalent forms with the LCD, then add/subtract the numerators. $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$.
Examples Subtraction needs an LCD: $\frac{7y}{12} \frac{5}{18}$ becomes $\frac{21y}{36} \frac{10}{36} = \frac{21y 10}{36}$.
Multiplication does not need an LCD: $\frac{5x}{6} \cdot \frac{3}{10} = \frac{15x}{60}$, which simplifies to $\frac{x}{4}$ by removing common factors.
Common Questions
Do you need a common denominator to multiply fractions?
No. Multiply numerators together and denominators together directly: (a/b) · (c/d) = ac/bd. No LCD is needed.
How do you divide fractions?
Multiply the first fraction by the reciprocal of the second: (a/b) ÷ (c/d) = (a/b) · (d/c).
When do you need an LCD for fractions?
Only for addition and subtraction. Convert both fractions to equivalent forms with the same denominator, then add or subtract numerators.
How do you subtract 7y/12 − 5/18?
Find the LCD of 12 and 18, which is 36. Convert: 21y/36 − 10/36 = (21y − 10)/36.
How do you multiply (5x/6) · (3/10)?
Multiply numerators and denominators: 15x/60, which simplifies by cancelling common factors to x/4.
What is the most common fraction operation mistake?
Finding an LCD when multiplying — it is unnecessary. LCD is only required for addition and subtraction.