Let W,X,Y and Z be four different digits selected from the set \ 1,2,3,4,5,6,7,8,9\. If the sum (W/X) + (Y/Z) is to be as small as possible, then (W/X) + (Y/Z) must equal

AMC 8 1994 Problem 20

AMC8 1994 年第 20 题

比与比例★★★☆☆

Let W,X,Y and

Z

be four different digits selected from the set

\{ 1,2,3,4,5,6,7,8,9\}.

If the sum

\dfrac{W}{X} + \dfrac{Y}{Z}

is to be as small as possible, then

\dfrac{W}{X} + \dfrac{Y}{Z}

must equal

设 W,X,Y 和

Z

是从集合

\{1,2,3,4,5,6,7,8,9\}

中选出的四个不同数字。

若要和

\dfrac{W}{X} + \dfrac{Y}{Z}

尽可能小，则

\dfrac{W}{X} + \dfrac{Y}{Z}

必须等于

正确答案 · Correct Answer

Solution 1

\displaystyle \frac{W}{X} + \frac{Y}{Z} = \frac{WZ+XY}{XZ}

Small fractions have small numerators and large denominators. To maximize the denominator, let X=8 and Z=9.

\displaystyle \frac{9W+8Y}{72}

To minimize the numerator, let W=1 and Y=2.

\displaystyle \frac{9+16}{72} = \textbf{(D)}\ \frac{25}{72}

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