分数の逆数と比例式（１２） - 丿乀庵【へつぽつあん】

●既約分数が $\ \frac{3}{4}\$ になる分数の逆数の例

（既約分数の分母と分子に色をつけて区別できるようにした）

① $\ \frac{3}{4}$ と、その逆数の掛け算

$\frac{3}{4} × \frac{4}{3} = \frac {\color{red}{3}×1} {\color{green}{4}×1} × \frac {\color{green}{4}×1} {\color{red}{3}×1} = \frac{12}{12} = 1$

$\frac{3}{4} × \frac{8}{6} = \frac {\color{red}{3}×1} {\color{green}{4}×1} × \frac {\color{green}{4}×2} {\color{red}{3}×2} = \frac{24}{24} = 1$

$\frac{3}{4} × \frac{12}{9} = \frac {\color{red}{3}×1} {\color{green}{4}×1} × \frac {\color{green}{4}×3} {\color{red}{3}×3} = \frac{36}{36} = 1$

$\frac{3}{4} × \frac{16}{12} = \frac {\color{red}{3}×1} {\color{green}{4}×1} × \frac {\color{green}{4}×4} {\color{red}{3}×4} = \frac{48}{48} = 1$

$\frac{3}{4} × \frac{20}{15} = \frac {\color{red}{3}×1} {\color{green}{4}×1} × \frac {\color{green}{4}×5} {\color{red}{3}×5} = \frac{60}{60} = 1$

② $\ \frac{6}{8}$ と、その逆数の掛け算

$\frac{6}{8} × \frac{4}{3} = \frac {\color{red}{3}×2} {\color{green}{4}×2} × \frac {\color{green}{4}×1} {\color{red}{3}×1} = \frac{24}{24} = 1$

$\frac{6}{8} × \frac{8}{6} = \frac {\color{red}{3}×2} {\color{green}{4}×2} × \frac {\color{green}{4}×2} {\color{red}{3}×2} = \frac{48}{48} = 1$

$\frac{6}{8} × \frac{12}{9} = \frac {\color{red}{3}×2} {\color{green}{4}×2} × \frac {\color{green}{4}×3} {\color{red}{3}×3} = \frac{72}{72} = 1$

$\frac{6}{8} × \frac{16}{12} = \frac {\color{red}{3}×2} {\color{green}{4}×2} × \frac {\color{green}{4}×4} {\color{red}{3}×4} = \frac{96}{96} = 1$

$\frac{6}{8} × \frac{20}{15} = \frac {\color{red}{3}×2} {\color{green}{4}×2} × \frac {\color{green}{4}×5} {\color{red}{3}×5} = \frac{120}{120} = 1$

③ $\ \frac{9}{12}$ と、その逆数の掛け算

$\frac{9}{12} × \frac{4}{3} = \frac {\color{red}{3}×3} {\color{green}{4}×3} × \frac {\color{green}{4}×1} {\color{red}{3}×1} = \frac{36}{36} = 1$

$\frac{9}{12} × \frac{8}{6} = \frac {\color{red}{3}×3} {\color{green}{4}×3} × \frac {\color{green}{4}×2} {\color{red}{3}×2} = \frac{72}{72} = 1$

$\frac{9}{12} × \frac{12}{9} = \frac {\color{red}{3}×3} {\color{green}{4}×3} × \frac {\color{green}{4}×3} {\color{red}{3}×3} = \frac{108}{108} = 1$

$\frac{9}{12} × \frac{16}{12} = \frac {\color{red}{3}×3} {\color{green}{4}×3} × \frac {\color{green}{4}×4} {\color{red}{3}×4} = \frac{144}{144} = 1$

$\frac{9}{12} × \frac{20}{15} = \frac {\color{red}{3}×3} {\color{green}{4}×3} × \frac {\color{green}{4}×5} {\color{red}{3}×5} = \frac{180}{180} = 1$

④ $\ \frac{12}{16}$ と、その逆数の掛け算

$\frac{12}{16} × \frac{4}{3} = \frac {\color{red}{3}×4} {\color{green}{4}×4} × \frac {\color{green}{4}×1} {\color{red}{3}×1} = \frac{48}{48} = 1$

$\frac{12}{16} × \frac{8}{6} = \frac {\color{red}{3}×4} {\color{green}{4}×4} × \frac {\color{green}{4}×2} {\color{red}{3}×2} = \frac{96}{96} = 1$

$\frac{12}{16} × \frac{12}{9} = \frac {\color{red}{3}×4} {\color{green}{4}×4} × \frac {\color{green}{4}×3} {\color{red}{3}×3} = \frac{144}{144} = 1$

$\frac{12}{16} × \frac{16}{12} = \frac {\color{red}{3}×4} {\color{green}{4}×4} × \frac {\color{green}{4}×4} {\color{red}{3}×4} = \frac{192}{192} = 1$

$\frac{12}{16} × \frac{20}{15} = \frac {\color{red}{3}×4} {\color{green}{4}×4} × \frac {\color{green}{4}×5} {\color{red}{3}×5} = \frac{240}{240} = 1$

⑤ $\ \frac{15}{20}$ と、その逆数の掛け算

$\frac{15}{20} × \frac{4}{3} = \frac {\color{red}{3}×5} {\color{green}{4}×5} × \frac {\color{green}{4}×1} {\color{red}{3}×1} = \frac{60}{60} = 1$

$\frac{15}{20} × \frac{8}{6} = \frac {\color{red}{3}×5} {\color{green}{4}×5} × \frac {\color{green}{4}×2} {\color{red}{3}×2} = \frac{120}{120} = 1$

$\frac{15}{20} × \frac{12}{9} = \frac {\color{red}{3}×5} {\color{green}{4}×5} × \frac {\color{green}{4}×3} {\color{red}{3}×3} = \frac{180}{180} = 1$

$\frac{15}{20} × \frac{16}{12} = \frac {\color{red}{3}×5} {\color{green}{4}×5} × \frac {\color{green}{4}×4} {\color{red}{3}×4} = \frac{240}{240} = 1$

$\frac{15}{20} × \frac{20}{15} = \frac {\color{red}{3}×5} {\color{green}{4}×5} × \frac {\color{green}{4}×5} {\color{red}{3}×5} = \frac{300}{300} = 1$