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prealgebra intermediate

Problem

Consider the operation "minus the reciprocal of," defined by

a ⋄ b = a - \frac{1}{b}

. What is

((1 ⋄ 2) ⋄ 3) - (1 ⋄ (2 ⋄ 3))

?

(A)

- \frac{7}{30}

(B)

- \frac{1}{6}

(C)

0

(D)

\frac{1}{6}

Solution

1 ⋄ 2 = 1 - \frac{1}{2} = \frac{1}{2}

, so

(1 ⋄ 2) ⋄ 3 = \frac{1}{2} ⋄ 3 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

. Also,

2 ⋄ 3 = 2 - \frac{1}{3} = \frac{5}{3}

, so

1 ⋄ (2 ⋄ 3) = 1 - \frac{1}{5/3} = 1 - \frac{3}{5} = \frac{2}{5}

. Thus,

((1 ⋄ 2) ⋄ 3) - (1 ⋄ (2 ⋄ 3)) = \frac{1}{6} - \frac{2}{5} = - \frac{7}{30}

Final answer

A