AMC10 2017 A

AMC10 2017 A · Q5

AMC10 2017 A · Q5. It mainly tests Linear equations, Rational expressions.

The sum of two nonzero real numbers is 4 times their product. What is the sum of the reciprocals of the two numbers?

两个非零实数的和是它们积的 4 倍。这两个数的倒数之和是多少？

(A) 1 1

(B) 2 2

(D) 8 8

(E) 12 12

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let the two numbers be \(x\) and \(y\). Then \(x + y = 4xy\). Dividing this equation by \(xy\) gives \[\frac{1}{y} + \frac{1}{x} = 4.\]One such pair of numbers is \(x = \frac{1}{3}\), \(y = 1\).

设两个数为 \(x\) 和 \(y\)。则 \(x + y = 4xy\)。将此方程除以 \(xy\) 得 \[\frac{1}{y} + \frac{1}{x} = 4.\] 一个这样的数对是 \(x = \frac{1}{3}\)，\(y = 1\)。

Topics