Cross-Multiplication
Fractions & DecimalsCross-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: the top-left times the bottom-right equals the top-right times the bottom-left. It is called "cross" because you multiply in an X pattern.
Example
Solve: $3/4 = x/12$. Cross-multiply: $3 \times 12 = 4 \times x$. $36 = 4x$. $x = 9$. Check: $3/4 = 9/12 = 3/4$. Correct!
Key Insight
Cross-multiplication only works when two fractions are set equal to each other (a proportion). It transforms a fraction equation into a simpler multiplication equation by clearing all denominators at once.
Definition
Cross-multiplication states that $a/b = c/d$ iff $ad = bc$ (where $b, d \neq 0$). This is proved by multiplying both sides of $a/b = c/d$ by $bd$: $ad = bc$. It can be used to solve for an unknown in a proportion, compare two fractions ($a/b < c/d$ iff $ad < bc$ when $b, d > 0$), or verify that two ratios are equivalent.
Example
Is $7/11 > 8/13$? Cross-multiply: $7 \times 13 = 91$ and $11 \times 8 = 88$. Since $91 > 88$, yes, $7/11 > 8/13$. Solve: $5/(x+2) = 3/7$. Cross-multiply: $35 = 3(x+2)$. $35 = 3x+6$. $3x = 29$. $x = 29/3$.
Key Insight
Cross-multiplication is the algebraic form of the "same denominators" comparison trick. Multiplying both sides by the product of denominators ($bd$) converts the proportion to a linear equation, making it solvable by standard methods.
Definition
Cross-multiplication is the application of the field axioms to clear denominators: $a/b = c/d$ is equivalent to $ad = bc$ in any field (or integral domain, where cancellation is valid). In the integers, this is the definition of the equivalence relation on pairs that constructs $\mathbb{Q}$. In algebraic geometry, cross-multiplication appears in the definition of rational maps between varieties.
Example
In solving rational equations: $1/(x-1) + 1/(x+1) = 4/(x^2-1)$. Note $x^2-1 = (x-1)(x+1)$. Multiply through by $(x-1)(x+1)$: $(x+1) + (x-1) = 4$. $2x = 4$. $x = 2$. Check: $1/1 + 1/3 = 4/3$. Indeed $4/3 = 4/3$.
Key Insight
Cross-multiplication is an instance of the more general principle of "clearing denominators," which transforms rational (fractional) equations into polynomial equations. This technique underlies partial fraction decomposition, residue computation in complex analysis, and rational interpolation in numerical methods.