【計算の工夫】部分分数分解で掛け算を引き算にするキセル算が楽しい

問題演習コーナー

【問題】次の計算をしなさい。

(1) \(\frac{3}{3×6}\)＋ \(\frac{3}{6×9}\)＋\(\frac{3}{9×12}\)＋\(\frac{3}{12×15}\)＋\(\frac{3}{15×18}\)

(2) \(\frac{1}{1×3}\)＋ \(\frac{1}{2×4}\)＋\(\frac{1}{3×5}\)＋\(\frac{1}{4×6}\)＋\(\frac{1}{5×7}\)＋\(\frac{1}{6×8}\)

(3) \(\frac{1}{1×2×3}\)＋ \(\frac{1}{2×3×4}\)＋\(\frac{1}{3×4×5}\)＋…＋\(\frac{1}{7×8×9}\)＋\(\frac{1}{8×9×10}\)

(1) \(\frac{1}{3}－\frac{1}{6}\)＝\(\frac{3}{3×6}\)で分子が3になるので、\(\frac{3}{3×6}\)の部分分数分解では、引き算の式\(\frac{1}{3}－\frac{1}{6}\)に何もかけなくていいことがわかります。

\(\left(\frac{1}{3}－\frac{1}{6}\right)\)＋\(\left(\frac{1}{6}－\frac{1}{9}\right)\)＋\(\left(\frac{1}{9}－\frac{1}{12}\right)\)＋\(\left(\frac{1}{12}－\frac{1}{15}\right)\)＋\(\left(\frac{1}{15}－\frac{1}{18}\right)\)＝\(\frac{1}{3}\)－\(\frac{1}{18}\)＝\(\underline{\frac{5}{18}}\)

(2) \(\frac{1}{1}－\frac{1}{3}\)＝\(\frac{2}{1×3}\)で分子が2になるので、\(\frac{1}{1×3}\)の部分分数分解では、引き算の式\(\frac{1}{1}－\frac{1}{3}\)に\(\frac{1}{2}\)をかければいいことがわかります。

\(\frac{1}{2}\)×\(\left(\frac{1}{1}－\frac{1}{3}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{2}－\frac{1}{4}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{3}－\frac{1}{5}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{4}－\frac{1}{6}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{5}－\frac{1}{7}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{6}－\frac{1}{8}\right)\)

＝\(\frac{1}{2}\)×\(\left(\frac{1}{1}＋\frac{1}{2}－\frac{1}{7}－\frac{1}{8}\right)\)

＝\(\frac{1}{2}\)×\(\left(\frac{56}{56}＋\frac{28}{56}－\frac{8}{56}－\frac{7}{56}\right)\)

＝\(\frac{1}{2}\)×\(\frac{69}{56}\)

＝\(\underline{\frac{69}{112}}\)

(3) \(\frac{1}{1×2}－\frac{1}{2×3}\)＝\(\frac{2}{1×2×3}\)で分子が2になるので、\(\frac{1}{1×2×3}\)の部分分数分解では、引き算の式\(\frac{1}{1×2}－\frac{1}{2×3}\)に\(\frac{1}{2}\)をかければいいことがわかります。

\(\frac{1}{2}\)×\(\left(\frac{1}{1×2}－\frac{1}{2×3}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{2×3}－\frac{1}{3×4}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{3×4}－\frac{1}{4×5}\right)\)＋…＋\(\frac{1}{2}\)×\(\left(\frac{1}{7×8}－\frac{1}{8×9}\right)\)＋\(\frac{1}{2}\)×\(\left(\frac{1}{8×9}－\frac{1}{9×10}\right)\)

＝\(\frac{1}{2}\)×\(\left(\frac{1}{1×2}－\frac{1}{9×10}\right)\)

＝\(\frac{1}{2}\)×\(\left(\frac{45}{90}－\frac{1}{90}\right)\)

＝\(\frac{1}{2}\)×\(\frac{44}{90}\)

＝\(\underline{\frac{11}{45}}\)

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