Cross-Multiplication
Cross-multiplication is a method for solving proportions by multiplying the numerator of each fraction by the denominator of the other and setting the products equal.
Formula
\text{if } \frac{a}{b} = \frac{c}{d}, \text{ then } ad = bc
Definition
Cross-multiplication is a shortcut for solving proportions: when two fractions are equal ($a/b = c/d$), you can multiply diagonally, top-left times bottom-right equals top-right times bottom-left, an "X" pattern that only works when two fractions are set equal to each other. Formally, this states $a/b = c/d$ iff $ad = bc$ (where $b, d \neq 0$), proved by multiplying both sides of $a/b = c/d$ by $bd$; it can solve for an unknown in a proportion or compare two fractions ($a/b < c/d$ iff $ad < bc$ when $b, d > 0$). More abstractly, cross-multiplication is the application of field axioms to clear denominators, valid in any field or integral domain where cancellation holds, and it appears in the definition of rational maps between algebraic varieties.
Example
Solving $3/4 = x/12$: cross-multiply to get $3 \times 12 = 4 \times x$, so $36 = 4x$ and $x = 9$ (check: $3/4 = 9/12 = 3/4$). Comparing $7/11$ and $8/13$: $7 \times 13 = 91$ and $11 \times 8 = 88$, and since $91 > 88$, $7/11 > 8/13$. Clearing denominators solves rational equations too: $1/(x-1) + 1/(x+1) = 4/(x^2-1)$, noting $x^2-1 = (x-1)(x+1)$, multiply through by that product to get $(x+1) + (x-1) = 4$, so $2x = 4$ and $x = 2$ (check: $1/1 + 1/3 = 4/3$).
Key Insight
Cross-multiplication transforms a fraction equation into a simpler multiplication equation by clearing all denominators at once, and it is really just the algebraic form of the "same denominators" comparison trick: multiplying both sides by the product of denominators converts the proportion into a linear equation solvable by standard methods. This principle of "clearing denominators" is far more general, underlying partial fraction decomposition, residue computation in complex analysis, and rational interpolation in numerical methods.