Cross-multiplication
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.
The method is also known as the "cross your heart" method because a heart can be drawn to remember which things to multiply together and the lines resemble a heart outline.
Given an equation like:
(where b and d are not zero), one can cross-multiply to get:
In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.
In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over.
The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation:
we can multiply the terms on each side by the same number and the terms will remain equal. Therefore, if we multiply the fraction on each side by the product of the denominators of both sides—bd—we get:
We can reduce the fractions to lowest terms by noting that the two occurrences of on the left-hand side cancel, as do the two occurrences of d on the right-hand side, leaving:
and we can divide both sides of the equation by any of the elements—in this case we will use d—getting:
Another justification of cross-multiplication is as follows. Starting with the given equation:
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