2017 AMC 12A 第 2 题

Question

两个非零实数的和等于它们乘积的

4

倍。这两个数的倒数之和是多少？

The sum of two nonzero real numbers is

4

times their product. What is the sum of the reciprocals of the two numbers?

1

2

4

8

12

倒数之和为

\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{x+y}{xy}

。

The sum of reciprocals is

\dfrac{1}{x}+\dfrac{1}{y}=\dfrac{x+y}{xy}

将方程

x+y=4xy

两边同时除以

xy

。

Divide the equation

x+y=4xy

by

xy

设这两个数为

x

和

y

，则

x+y=4xy

。

两边同时除以

xy

，得到

\dfrac{x+y}{xy}=4\text{，}

而左边正是

\dfrac{1}{y}+\dfrac{1}{x}

。因此倒数之和为

4

。

所以正确答案是 C。

Let the numbers be

x

and

y,

so

x+y=4xy.

Dividing both sides by

xy

gives

\dfrac{x+y}{xy}=4,

and the left side is exactly

\dfrac{1}{y}+\dfrac{1}{x}.

So the sum of the reciprocals is

4.

Thus, the correct answer is C.

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