次の循環小数を分数で表す問題です。 (1) $0.\dot{5}$ (2) $0.1\dot{9}$ (3) $0.\dot{2}\dot{7}$

1. 問題の内容

次の循環小数を分数で表す問題です。

(1) $0.\dot{5}$

(2) $0.1\dot{9}$

(3) $0.\dot{2}\dot{7}$

2. 解き方の手順

(1) $x = 0.\dot{5}$ とおくと、$x = 0.555...$

$10x = 5.555...$

$10x - x = 5.555... - 0.555...$

$9x = 5$

$x = \frac{5}{9}$

(2) $x = 0.1\dot{9}$ とおくと、$x = 0.1999...$

$10x = 1.999...$

$100x = 19.999...$

$100x - 10x = 19.999... - 1.999...$

$90x = 18$

$x = \frac{18}{90} = \frac{1}{5}$

(3) $x = 0.\dot{2}\dot{7}$ とおくと、$x = 0.272727...$

$100x = 27.272727...$

$100x - x = 27.272727... - 0.272727...$

$99x = 27$

$x = \frac{27}{99} = \frac{3}{11}$

3. 最終的な答え

(1) $\frac{5}{9}$

(2) $\frac{1}{5}$

(3) $\frac{3}{11}$