Products of Fractions
Products of Fractions teaches the rule for multiplying two fractions: multiply numerators together and denominators together, (a/b)(c/d) = ac/bd. This foundational operation is covered in Yoshiwara Elementary Algebra Chapter 8: Algebraic Fractions and applies to both numerical and algebraic fractions. Grade 6 students often simplify by canceling common factors before multiplying to make computation easier.
Key Concepts
Property To multiply two fractions, multiply their numerators together and multiply their denominators together. If $b \neq 0$ and $d \neq 0$, then $$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$$ To multiply algebraic fractions: 1. Factor each numerator and denominator completely. 2. If any factor appears in both a numerator and a denominator, divide out that factor. 3. Multiply the remaining factors of the numerator and the remaining factors of the denominator. 4. Reduce the product if necessary.
Examples To multiply $\frac{4}{5} \cdot \frac{2}{3}$, we calculate $\frac{4 \cdot 2}{5 \cdot 3} = \frac{8}{15}$. To multiply $\frac{5x}{6} \cdot \frac{9}{10x^2}$, we factor and cancel: $\frac{5x}{2 \cdot 3} \cdot \frac{3 \cdot 3}{2 \cdot 5x \cdot x} = \frac{3}{4x}$. To multiply $\frac{x+3}{2x 8} \cdot \frac{x 4}{x^2 9}$, we factor first: $\frac{x+3}{2(x 4)} \cdot \frac{x 4}{(x 3)(x+3)} = \frac{1}{2(x 3)}$.
Explanation Multiplying fractions means finding a part of a part. Multiply the tops (numerators) to get the new number of pieces, and multiply the bottoms (denominators) to find the new total size. Always cancel common factors first to simplify!
Common Questions
How do you multiply two fractions?
Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator: (a/b) × (c/d) = ac/bd.
Can you simplify before multiplying fractions?
Yes — canceling common factors before multiplying (cross-canceling) makes the numbers smaller and simplification easier. Always reduce the final answer to lowest terms.
How does multiplying algebraic fractions differ from numerical fractions?
The process is the same, but with variables. For example, (x/2)(3/y) = 3x/2y. Factor and cancel any common variable or numerical factors.
Where is products of fractions taught in Yoshiwara Elementary Algebra?
Products of Fractions is covered in Chapter 8: Algebraic Fractions of Yoshiwara Elementary Algebra.
What is the result when you multiply a fraction by its reciprocal?
Multiplying any nonzero fraction by its reciprocal always equals 1: (a/b)(b/a) = ab/ab = 1.