2020 AMC 10A 第 3 题

Question

假设

a\neq3

、

b\neq4

、

c\neq5

，下列表达式的最简值是多少？

\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}

Assuming

a\neq3,

b\neq4,

and

c\neq5,

what is the value in simplest form of the following expression?

\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}

-1

1

\displaystyle \frac{abc}{60}

\displaystyle \frac{1}{abc} - \frac{1}{60}

\displaystyle \frac{1}{60} - \frac{1}{abc}

每个分母都是对应分子因子的相反数。

Each denominator is the negative of a matching numerator factor

约分后数一数负号的个数。

Count the number of negative signs after cancellation

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将分母各因式改写为

5-c=-(c-5)

、

3-a=-(a-3)

和

4-b=-(b-4)

。原式化为

\dfrac{(a-3)(b-4)(c-5)}{-(a-3)(b-4)(c-5)}=-1

。正确答案是 A。

Rewrite the denominator factors as

5-c=-(c-5)

,

3-a=-(a-3)

, and

4-b=-(b-4)

. The expression becomes

\dfrac{(a-3)(b-4)(c-5)}{-(a-3)(b-4)(c-5)}=-1

. Thus, A is the correct answer.

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