2015 AMC 10B Problem 7Problem 7 of 25·Easier·AlgebraArithmeticConsider the operation “minus the reciprocal of,” defined by a⋄b=a−1b.a\diamond b=a-\frac{1}{b}.a⋄b=a−b1. What is ((1⋄2)⋄3)−(1⋄(2⋄3))?((1\diamond2)\diamond3)-(1\diamond(2\diamond3))?((1⋄2)⋄3)−(1⋄(2⋄3))?Answer choicesA−730 -\dfrac{7}{30} −3071B−16 -\dfrac{1}{6} −612C0 0 03D16 \dfrac{1}{6} 614E730 \dfrac{7}{30} 3075Submit answerStuck? Show hintsShow solutionSolution((1⋄2)⋄3)−(1⋄(2⋄3))=(12−13)−(1⋄53)=(12−13)−(1−35)=16−25=−730\begin{aligned} &((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) \\ &= \left(\dfrac 12- \dfrac 13\right) - \left(1 \diamond \dfrac 53\right)\\ &= \left(\dfrac 12 - \dfrac 13\right) - \left(1-\dfrac 35\right)\\ &= \dfrac 16 - \dfrac 25 \\&= -\dfrac{7}{30} \end{aligned}((1⋄2)⋄3)−(1⋄(2⋄3))=(21−31)−(1⋄35)=(21−31)−(1−53)=61−52=−307 Thus, the correct answer is A .AoPS wikiCopy problemTagged: custom operation · fraction