Subjects algebra

Infinite Decimal

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1. 题目说明:小马虎将 $2.36\overline{36}$ (表示 $2.363636\ldots$)误写成了 $2.36$,结果乘某数得出结果比正确答案少 $2.36$,求正确答案。 2. 设该某数为 $x$。正确的乘积为 $2.36\overline{36} \times x$,错误的乘积为 $2.36 \times x$。 3. 根据题意,有 $$2.36\overline{36} \times x - 2.36 \times x = 2.36.$$ 4. 化简左边: $$x (2.36\overline{36} - 2.36) = 2.36.$$ 5. 先计算 $2.36\overline{36} - 2.36$ : 设 $y = 2.36\overline{36}$,把无限循环数表示成分数。 6. 令 $y = 2.363636\ldots$ 7. 乘以 100: $$100y = 236.363636\ldots$$ 8. 将 $100y$ 减去 $y$: $$100y - y = 236.363636\ldots - 2.363636\ldots = 234.$$ 9. 整理得: $$99y = 234 \,\Rightarrow\, y = \frac{234}{99} = \frac{78}{33} = \frac{26}{11}.$$ 10. 计算差值: $$2.36\overline{36} - 2.36 = \frac{26}{11} - \frac{236}{100}.$$ 将 $2.36$ 转化为分数: $$2.36 = \frac{236}{100} = \frac{59}{25}.$$ 11. 通分减法: $$\frac{26}{11} - \frac{59}{25} = \frac{26 \times 25}{11 \times 25} - \frac{59 \times 11}{25 \times 11} = \frac{650}{275} - \frac{649}{275} = \frac{1}{275}.$$ 12. 因此, $$x \times \frac{1}{275} = 2.36 = \frac{59}{25}.$$ 13. 解 $x$: $$x = 2.36 \times 275 = \frac{59}{25} \times 275 = \frac{59 \times 275}{25} = 59 \times 11 = 649.$$ 14. 正确答案为: $$2.36\overline{36} \times x = y \times x = \frac{26}{11} \times 649 = 26 \times 59 = 1534.$$ 答:正确答案是 $1534$。