Question

两个正数的乘积为 $9$ 。其中一个数的倒数是另一个数倒数的 $4$ 倍。这两个数的和是多少？

The product of two positive numbers is $9.$ The reciprocal of one of these numbers is $4$ times the reciprocal of the other number. What is the sum of the two numbers?

$\dfrac{10}{3}$

$\dfrac{20}{3}$

$7$

$\dfrac{15}{2}$

$8$

答案：D

知识点：方程组分式方程

难度评级：1070

解答：

设两个数为 $x$ 和 $y$ ，满足 $xy = 9 \text{ and } \dfrac{1}{x} = \dfrac{4}{y}.$

由乘积条件得代入倒数条件，得到因为 $x$ 为正，所以 $y = \dfrac{9}{x}$ $\dfrac{1}{x} = \dfrac{4x}{9}.$ $x = \dfrac{3}{2},$

于是另一个数为 $y = 9 \div \dfrac{3}{2} = 6.$

两数之和为 $6 + \dfrac{3}{2} = \dfrac{15}{2}.$

所以正确答案是 D。

Let the two numbers be $x$ and $y$ such that $xy = 9 \text{ and } \dfrac{1}{x} = \dfrac{4}{y}.$

We get that $y = \dfrac{9}{x}$ $\dfrac{1}{x} = \dfrac{4x}{9}.$ $x = \dfrac{3}{2},$ since $x$ is positive.

Then $y = 9 \div \dfrac{3}{2} = 6.$

The desired sum is then $6 + \dfrac{3}{2} = \dfrac{15}{2}.$

Thus, D is the correct answer.

2012 AMC 10A 第 6 题

答案：D

难度评级：1070

解答：

其他年份的第 6 题