What is the value of 2a^-1+a^-12a when a=12?

2016 AMC 12B Problem 1

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What is the value of

\frac{2a^{-1}+\frac{a^{-1}}{2}}{a}

when

a=\tfrac12?

1

2

\dfrac52

10

20

a^{-1}=\frac{1}{a},

a^{-1}=2

when

a=\tfrac12

The numerator is

2a^{-1}+\tfrac12 a^{-1}=\tfrac52 a^{-1};

then divide by

a

With

a=\tfrac12,

we have

a^{-1}=2.

The numerator is

2\cdot2+\dfrac{2}{2}=4+1=5,

and dividing by

a=\tfrac12

gives

\dfrac{5}{\frac{1}{2}}=10.

Thus, the correct answer is D.