AMC10 2010 A

AMC10 2010 A · Q6

AMC10 2010 A · Q6. It mainly tests Linear equations, Fractions.

For positive numbers $x$ and $y$ the operation $\spadesuit(x, y)$ is defined as $\spadesuit(x, y) = x - 1/y$. What is $\spadesuit(2, \spadesuit(2, 2))$?

对于正数$x$和$y$，操作$\spadesuit(x, y)$定义为$\spadesuit(x, y) = x - 1/y$。求$\spadesuit(2, \spadesuit(2, 2))$的值。

(A) $\frac{2}{3}$ $\frac{2}{3}$

(B) 1 1

(D) $\frac{5}{3}$ $\frac{5}{3}$

(E) 2 2

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Note that $\spadesuit(2,2)=2-\frac{1}{2}=\frac{3}{2}$. Therefore \[ \spadesuit(2,\spadesuit(2,2))=\spadesuit\left(2,\frac{3}{2}\right)=2-\frac{2}{3}=\frac{4}{3}. \]

答案（C）：注意 $\spadesuit(2,2)=2-\frac{1}{2}=\frac{3}{2}$。因此 \[ \spadesuit(2,\spadesuit(2,2))=\spadesuit\left(2,\frac{3}{2}\right)=2-\frac{2}{3}=\frac{4}{3}. \]

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