問題は、次の循環小数を分数で表すことです。 (1) $0.\dot{7}$ (2) $3.\dot{7}\dot{2}$ (3) $1.\dot{2}1\dot{6}$ (4) $0.2\dot{4}\dot{6}$

1. 問題の内容

問題は、次の循環小数を分数で表すことです。

(1) $0.\dot{7}$

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

(3) $1.\dot{2}1\dot{6}$

(4) $0.2\dot{4}\dot{6}$

2. 解き方の手順

(1) $x = 0.\dot{7}$ とおく。

$10x = 7.\dot{7}$

$10x - x = 7.\dot{7} - 0.\dot{7} = 7$

$9x = 7$

$x = \frac{7}{9}$

(2) $x = 3.\dot{7}\dot{2}$ とおく。

$100x = 372.\dot{7}\dot{2}$

$100x - x = 372.\dot{7}\dot{2} - 3.\dot{7}\dot{2} = 369$

$99x = 369$

$x = \frac{369}{99} = \frac{41}{11}$

(3) $x = 1.\dot{2}1\dot{6}$ とおく。

$100x = 121.6\dot{2}1$

$1000x = 1216.\dot{2}1\dot{6}$

$1000x - x = 1216.\dot{2}1\dot{6} - 1.\dot{2}1\dot{6} = 1215$

$999x = 1215$

$x = \frac{1215}{999} = \frac{405}{333} = \frac{135}{111} = \frac{45}{37}$

(4) $x = 0.2\dot{4}\dot{6}$ とおく。

$10x = 2.\dot{4}\dot{6}$

$1000x = 246.\dot{4}\dot{6}$

$1000x - 10x = 246.\dot{4}\dot{6} - 2.\dot{4}\dot{6} = 244$

$990x = 244$

$x = \frac{244}{990} = \frac{122}{495}$

3. 最終的な答え

(1) $\frac{7}{9}$

(2) $\frac{41}{11}$

(3) $\frac{45}{37}$

(4) $\frac{122}{495}$