Wrong Answer - Rewrite Rational Number as a Fraction.

This number 2.962962 can be rational

$$x=2.962962$$ $$10x=29.62962$$ $$100x=296.2962$$ $$1000x=2962.962$$ $$1000x-10x=\frac{990x}{990}=\frac{2933}{990}$$

why is this wrong? That way of getting the answer is how I was said to do it

Comment: $$1000x-x=\frac{999x}{999}=\frac{2960}{999}=?$$

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The last line is wrong. If the decimal part is supposed to be repeating, I imagine it should read $1000x - x = 999x = 2960$. – Qiaochu Yuan Apr 2 '13 at 8:54
Your solution makes very less sense. You have made many direct mistakes. Check it once again. – lsp Apr 2 '13 at 8:55
@QiaochuYuan That can't be right because the answer has to be rational – user1838781 Apr 2 '13 at 8:56
@user: Can you explain your reasoning in that last comment? – Hurkyl Apr 2 '13 at 9:02
@Hurkyl it has to be a fraction = no decimals – user1838781 Apr 2 '13 at 9:06

$$1000x -x = 2962.962962\ldots - 2.962962\ldots = 2960.$$ The fractional part cancels out completely.

Then $$x = \frac{2960}{999} = \frac{37\cdot80}{37\cdot27} = \frac{80}{27}.$$

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I assume you meant a repeating decimal like $2.\overline{962}$ (i.e. $2.9629629\ldots$) rather than one that terminates like you write.

Your work at the end is somewhat confused; I can't tell what you were trying to do. But the calculation of $1000x - 10x$ yields

$$\begin{matrix} 2&9&\not 6^5&{}^1 2&.&9&\not6^5&{}^1 2&9&\not6^5&{}^12&... \\ & & 2&9&.&6&2&9&6&2&9&... \\\hline \\ 2 & 9 & 3 & 3 & . & 3 & 3 & 3 & 3 & 3 & 3 & \cdots \end{matrix}$$

(I hope that is how they still notate subtraction these days) and so you have

$$1000 x - 10 x = 2933 + \frac{1}{3}$$

and

$$1000 x - 10 x = 990 x$$

and so we've derived

$$990 x = 2933 + \frac{1}{3}$$

Of course, it would have been easier to compare $1000x$ to $x$....

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Hint: there's a very easy way to transform periodic into decimals without any calculation. Here's an example: $$0.123\overline{4567}=\frac{1234567-123}{9999000}.$$ In your case it's just $$2\frac{962}{999}.$$

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The number $2.962962$ certainly is rational, because $$2.962962 = \frac {2962962}{1000000}$$

BUT... if you mean $2.962962\ldots$, supposed to mean $2.962\overline{962} = 2.\overline{962}$, then $$1000\cdot 2.\overline{962} = 2962.\overline{962}$$ so $$1000\cdot 2.\overline{962} - 2.\overline{962} = 2960$$ thus $$999\cdot 2.\overline{962} = 2960$$ and $$2.\overline{962} = \frac {2960}{999}$$

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