Fundamental Principle of Fractions
The Fundamental Principle of Fractions states that multiplying or dividing both numerator and denominator by the same nonzero value produces an equivalent fraction: (a·c)/(b·c) = a/b. Covered in Yoshiwara Elementary Algebra Chapter 8: Algebraic Fractions, this principle is the foundation for simplifying fractions, building equivalent fractions, and finding common denominators. Grade 6 students apply it both to numerical fractions and algebraic rational expressions.
Key Concepts
Property We can multiply or divide the numerator and denominator of a fraction by the same nonzero factor, and the new fraction will be equivalent to the old one. $$\frac{a \cdot c}{b \cdot c} = \frac{a}{b} \quad \text{if } b, c \neq 0$$.
Examples We can reduce $\frac{18}{30}$ by noting $\frac{18}{30} = \frac{6 \cdot 3}{6 \cdot 5}$. We divide out the common factor 6 to get $\frac{3}{5}$.
The fraction $\frac{5x^2}{15x}$ can be written as $\frac{5 \cdot x \cdot x}{3 \cdot 5 \cdot x}$. Canceling the common factors of $5$ and $x$ gives $\frac{x}{3}$.
Common Questions
What is the fundamental principle of fractions?
Multiplying or dividing both the numerator and denominator by the same nonzero number gives an equivalent fraction. The value of the fraction does not change.
How do you use the fundamental principle to simplify fractions?
Divide both numerator and denominator by their common factors until no common factors remain. The result is the fraction in lowest terms.
How do you use the fundamental principle to build equivalent fractions?
Multiply both numerator and denominator by the same number to create an equivalent fraction with a larger denominator, useful for adding unlike fractions.
Where is the fundamental principle of fractions in Yoshiwara Elementary Algebra?
It is covered in Chapter 8: Algebraic Fractions of Yoshiwara Elementary Algebra.
Does the fundamental principle apply to algebraic fractions?
Yes. You can factor and cancel common algebraic factors from the numerator and denominator to simplify rational expressions.