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 $1534$。
Infinite Decimal
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