AMC10 2012 A

AMC10 2012 A · Q6

AMC10 2012 A · Q6. It mainly tests Linear equations, Fractions.

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?

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

(A) 10 10

(B) \frac{20}{3} \frac{20}{3}

(D) \frac{15}{2} \frac{15}{2}

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let $x > 0$ be the first number, and let $y > 0$ be the second number. The first statement implies $xy = 9$. The second statement implies $\frac{1}{x} = \frac{4}{y}$, so $y = 4x$. Substitution yields $x \cdot (4x) = 9$, so $x = \sqrt{9/4} = 3/2$. Therefore $x + y = 3/2 + 4 \cdot 3/2 = 15/2$.

设第一个数为$x > 0$，第二个数为$y > 0$。第一个条件给出$xy = 9$。第二个条件给出$\frac{1}{x} = \frac{4}{y}$，所以$y = 4x$。代入得$x \cdot (4x) = 9$，所以$x = \sqrt{9/4} = 3/2$。因此$x + y = 3/2 + 4 \cdot 3/2 = 15/2$。

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