Quod Erat Demonstrandum

2013/06/03

old Q in old pic

Filed under: Fun,Junior Form Mathematics — johnmayhk @ 3:52 下午

johnmayhk-math-from-old-pic

學校舊照,相信是七十年代某初中數學堂,隱約見黑(綠?)板上寫著某題:

If y=\frac{1}{1+x} , express \frac{x^2-2x+1}{x^2-x} in terms of y.

(希望沒有看錯)

最下一行似乎正在回答該問題,應該是未寫完吧,但方法似乎頗麻煩(但我始終看不清,不能定奪)。無論如何,初中同學可以試試這幾十年前的數。

Hint:

\frac{x^2-2x+1}{x^2-x}

=\frac{(x-1)^2}{x(x-1)}

=\frac{x-1}{x} for x\ne 1

=1-\frac{1}{x}

By the given,

y=\frac{1}{1+x}

\Rightarrow \frac{1}{y}=1+x

\Rightarrow \frac{1}{x}=\frac{y}{1-y}

Hence, the answer is \frac{1-2y}{1-y}.

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