Multiplying Fractions
To multiply fractions, multiply the numerators together and the denominators together, then simplify the result.
Formula
\left(\frac{a}{b}\right) \times \left(\frac{c}{d}\right) = \frac{ac}{bd}
Definition
To multiply two fractions, multiply the top numbers together for the new top and the bottom numbers together for the new bottom, no common denominator needed, just multiply straight across. The product of $a/b$ and $c/d$ is $(ac)/(bd)$, and it helps to simplify first by canceling any factor common to a numerator and any denominator (cross-cancellation), avoiding large numbers and reducing the need to simplify afterward. Formally, multiplication in $\mathbb{Q}$ is defined by $[a/b] \times [c/d] = [ac/bd]$, and verifying this is well-defined regardless of representative, combined with the additive structure, is what makes $\mathbb{Q}$ a field.
Example
$2/3 \times 3/4 = (2 \times 3)/(3 \times 4) = 6/12 = 1/2$; you can also cross-cancel first, the $3$s cancel, giving $2/4 = 1/2$ directly. For $(7/15) \times (10/21)$: $7$ and $21$ share factor $7$ ($7/7=1$, $21/7=3$), and $10$ and $15$ share factor $5$ ($10/5=2$, $15/5=3$), so the simplified product is $(1 \times 2)/(3 \times 3) = 2/9$. For rational functions in $\mathbb{C}(x)$: $(x^2-1)/(x+3) \times (x+3)/(x-1) = (x^2-1)/(x-1) = (x+1)(x-1)/(x-1) = x+1$ (for $x \neq 1$ or $-3$), where cancellation requires care about where the original functions are defined.
Key Insight
"Of" in math means multiply: "$1/2$ of $3/4$" $= 1/2 \times 3/4 = 3/8$, and multiplying fractions between $0$ and $1$ always gives a smaller result than either fraction, fractions of fractions get smaller. Multiplying fractions is the "easiest" fraction operation algorithmically, no common denominator required, the real difficulty is remembering to simplify, and cross-cancellation is just applying the associative and commutative laws to rearrange factors before multiplying. The commutativity and associativity of fraction multiplication follow from the same properties for integers, which is the power of building $\mathbb{Q}$ from $\mathbb{Z}$: algebraic laws are inherited, not re-proved from scratch.