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 occasionally known as the "cross your heart" method because lines resembling a heart outline can be drawn to remember which things to multiply together. 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 b 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 multiply by d/d = 1 on the left and by b/b = 1 on the right, getting and so Cancel the common denominator bd = db, leaving Each step in these procedures is based on a single, fundamental property of equations. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students. This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction.