2016 AMC 10B Problem 1

Since $a = \tfrac12$, its reciprocal is $a^{-1} = 2$. Substitute into the numerator:
$$2a^{-1} + \frac{a^{-1}}{2} = 2\cdot 2 + \frac{2}{2} = 4 + 1 = 5.$$
Now divide by $a = \tfrac12$. Dividing by one half is the same as multiplying by $2$:
$$\frac{5}{1/2} = 5 \cdot 2 = 10.$$

The answer is $\boxed{\textbf{(D)}\ 10}$.

A negative exponent is just a reciprocal, so $a^{-1}$ with $a = \tfrac12$ is the friendly number $2$. Substituting numbers immediately, rather than simplifying symbolically first, keeps a one-line problem at one line. Remember that dividing by a fraction less than $1$ makes the result larger.

Simplify symbolically first: the numerator is $\tfrac{5}{2}a^{-1}$, so the whole expression is $\tfrac{5}{2}a^{-2} = \tfrac{5}{2}\cdot 4 = 10$.

• Dividing by $a = \tfrac12$ as if it meant multiplying by $\tfrac12$, which gives $\tfrac52$ (choice C).
• Reading $a^{-1}$ as $-a$ or as $\tfrac12$ itself instead of its reciprocal $2$.