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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Rule of three (mathematics) ： ウィキペディア英語版 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. Given an equation like: :$\backslash frac\; a\; b\; =\; \backslash frac\; c\; d$ (where $b$ and $d$ are not zero), one can cross-multiply to get: :$ad\; =\; bc\; \backslash qquad\; \backslash mathrm\; \backslash qquad\; a\; =\; \backslash frac\; d.$ In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles. == Procedure == 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. :$\backslash frac\; a\; b\; \backslash nwarrow\; \backslash frac\; c\; d\; \backslash quad\; \backslash frac\; a\; b\; \backslash nearrow\; \backslash frac\; c\; d.$ The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation: :$\backslash frac\; a\; b\; =\; \backslash frac\; c\; d$ 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: :$\backslash frac\; a\; b\; \backslash times\; bd\; =\; \backslash frac\; c\; d\; \backslash times\; bd.$ 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: :$ad\; =\; bc$ and we can divide both sides of the equation by any of the elements—in this case we will use $d$—getting: :$a\; =\; \backslash frac\; d.$ Another justification of cross-multiplication is as follows. Starting with the given equation: :$\backslash frac\; a\; b\; =\; \backslash frac\; c\; d$ multiply by $\backslash frac\; d\; d\; =\; 1$ on the left and by $\backslash frac\; b\; b\; =\; 1$ on the right, getting: :$\backslash frac\; a\; b\; \backslash times\; \backslash frac\; d\; d\; =\; \backslash frac\; c\; d\; \backslash times\; \backslash frac\; b\; b$ and so: :$\backslash frac\; =\; \backslash frac\; .$ Cancel the common denominator $bd\; =\; db$, leaving: :$ad\; =\; cb.$ 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cross-multiplication」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.