1952 AMC 12 第 34 题

Question

一件商品的价格先增加

p\%

，随后新价格又降低

p\%

。若最终价格为一美元，则原价为：

The price of an article was increased

p\%.

Later the new price was decreased

p\%.

If the last price was one dollar, the original price was:

\dfrac{1-p^2}{200}

\dfrac{\sqrt{1-p^2}}{100}

一美元

One dollar

1-\dfrac{p^2}{10{,}000-p^2}

\dfrac{10{,}000}{10{,}000-p^2}

用乘法因子表示涨价和降价

Represent the increase and decrease by multiplicative factors

若原价为

P

，则

P(1+\frac{p}{100})(1-\frac{p}{100})=1

If the original price is

P,

then

P(1+\frac{p}{100})(1-\frac{p}{100})=1

若原价为

P

美元，则两次变价后的价格为

\begin{aligned} &P\left(1+\frac p{100}\right) \left(1-\frac p{100}\right)\\ &\qquad=P\left(1-\frac{p^2}{10{,}000}\right) \end{aligned}\text{。}

令该式等于

1

，得到

P=\frac{10{,}000}{10{,}000-p^2}\text{。}

因此，正确答案是 E。

If the original price is

P

dollars, then after both changes its price is

\begin{aligned} &P\left(1+\frac p{100}\right) \left(1-\frac p{100}\right)\\ &\qquad=P\left(1-\frac{p^2}{10{,}000}\right). \end{aligned}

Setting this equal to

1

gives

P=\frac{10{,}000}{10{,}000-p^2}.

Thus, the correct answer is E.

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