Simplify Fractions
Simplify Fractions is a foundational math skill from Openstax Intermediate Algebra 2E, Chapter 1: Foundations. A fraction a/b is simplified when the numerator and denominator share no common factors other than 1; students simplify by factoring both into primes and canceling all common factors using the Equivalent Fractions Property. This applies to numerical fractions as well as fractions with variable expressions.
Key Concepts
Property A fraction is written $\frac{a}{b}$, where $b \neq 0$ and $a$ is the numerator and $b$ is the denominator. A fraction represents parts of a whole. The denominator $b$ is the number of equal parts the whole has been divided into, and the numerator $a$ indicates how many parts are included.
Equivalent Fractions Property If $a$, $b$, and $c$ are numbers where $b \neq 0$, $c \neq 0$, then $\frac{a}{b} = \frac{a \cdot c}{b \cdot c}$ and $\frac{a \cdot c}{b \cdot c} = \frac{a}{b}$. A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. To simplify, factor the numerator and denominator into prime numbers, then divide out common factors.
Examples To simplify $ \frac{108}{63}$, we find the prime factors: $ \frac{2 \cdot 2 \cdot 3 \cdot 3 \cdot 3}{3 \cdot 3 \cdot 7}$. We cancel the common factors $3 \cdot 3$ to get $ \frac{2 \cdot 2 \cdot 3}{7} = \frac{12}{7}$.
Common Questions
How do you simplify a fraction?
Factor the numerator and denominator into prime factors, then cancel all factors that appear in both the numerator and the denominator.
What does it mean for a fraction to be fully simplified?
A fraction is fully simplified (in lowest terms) when its numerator and denominator share no common factors other than 1.
Can you cancel terms instead of factors when simplifying?
No, only factors (things that are multiplied) can be canceled; terms that are added or subtracted cannot be canceled across the fraction bar.
What textbook covers simplifying fractions?
Openstax Intermediate Algebra 2E, Chapter 1: Foundations covers simplifying numerical and algebraic fractions.