AMC10 2015 B

AMC10 2015 B · Q7

AMC10 2015 B · Q7. It mainly tests Manipulating equations, Fractions.

Consider the operation “minus the reciprocal of,” defined by $a \diamond b = a - \frac{1}{b}$. What is $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$?

考虑操作“减去倒数”，定义为 $a \diamond b = a - \frac{1}{b}$。求 $((1 \diamond 2) \diamond 3) - (1 \diamond (2 \diamond 3))$ 的值？

(A) $-\frac{7}{30}$ $-\frac{7}{30}$

(B) $-\frac{1}{6}$ $-\frac{1}{6}$

(D) $\frac{1}{6}$ $\frac{1}{6}$

(E) $\frac{7}{30}$ $\frac{7}{30}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): $((1\diamond 2)\diamond 3)-(1\diamond (2\diamond 3))=\left(\left(\left(1-\frac{1}{2}\right)-\frac{1}{3}\right)-\left(1-\left(2-\frac{1}{3}\right)^{-1}\right)\right)$ $=\frac{1}{6}-\left(1-\frac{3}{5}\right)=\frac{1}{6}-\frac{2}{5}=-\frac{7}{30}$

答案（A）： $((1\diamond 2)\diamond 3)-(1\diamond (2\diamond 3))=\left(\left(\left(1-\frac{1}{2}\right)-\frac{1}{3}\right)-\left(1-\left(2-\frac{1}{3}\right)^{-1}\right)\right)$ $=\frac{1}{6}-\left(1-\frac{3}{5}\right)=\frac{1}{6}-\frac{2}{5}=-\frac{7}{30}$

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