Multiplying Fractions
Fractions & DecimalsTo 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 to get the new top, and multiply the bottom numbers together to get the new bottom. No common denominator needed - just multiply straight across.
Example
$2/3 \times 3/4 = (2 \times 3)/(3 \times 4) = 6/12 = 1/2$. You can also simplify before multiplying (cross-cancel): $2/3 \times 3/4$ - the $3$s cancel, giving $2/4 = 1/2$.
Key Insight
"Of" in math means multiply. "$1/2$ of $3/4$" $= 1/2 \times 3/4 = 3/8$. Multiplying fractions always gives you a smaller result than either fraction (when both are between $0$ and $1$) - fractions of fractions get smaller.
Definition
The product of $a/b$ and $c/d$ is $(ac)/(bd)$. Before multiplying, simplify by canceling any factor common to a numerator and any denominator (cross-cancellation). This avoids working with large numbers and reduces the need to simplify the final answer.
Example
$(7/15) \times (10/21)$. Cross-cancel: $7$ and $21$ share factor $7$ ($7/7=1$, $21/7=3$); $10$ and $15$ share factor $5$ ($10/5=2$, $15/5=3$). Simplified: $(1 \times 2)/(3 \times 3) = 2/9$.
Key Insight
Multiplying fractions is the "easiest" fraction operation algorithmically - no common denominator required. The difficulty is remembering to simplify. Cross-cancellation is just applying the associative and commutative laws to rearrange factors before multiplying.
Definition
Multiplication in $\mathbb{Q}$ is defined by $[a/b] \times [c/d] = [ac/bd]$. Well-definedness: if $(a,b) \sim (a',b')$ and $(c,d) \sim (c',d')$, then $ab' = a'b$ and $cd' = c'd$, so $(ac)(b'd') = (ab')(cd') = (a'b)(c'd) = (a'c')(bd)$, confirming $(ac,bd) \sim (a'c',b'd')$. Combined with additive structure, this makes $\mathbb{Q}$ a field.
Example
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 $-3$). Cancellation requires care about where the original functions are defined.
Key Insight
The commutativity and associativity of fraction multiplication follow from the same properties for integers. This is the power of building $\mathbb{Q}$ from $\mathbb{Z}$: algebraic laws are inherited, not re-proved from scratch.