2016 AMC 10B Problem 1

The expression looks messy, but $a$ is a friendly number, so put numbers in right away.

1. A power of $-1$ means "one over": $a^{-1} = \frac{1}{a}$. With $a = \frac12$, that is $\frac{1}{1/2} = 2$.
2. Top of the fraction: $2a^{-1} = 2 \cdot 2 = 4$, and $\frac{a^{-1}}{2} = \frac{2}{2} = 1$. So the top is $4 + 1 = 5$.
3. Bottom of the fraction: $a = \frac12$.
4. Dividing by $\frac12$ means asking "how many halves fit into $5$?" The answer is $10$, because each whole holds two halves. So
$$\frac{5}{1/2} = 5 \cdot 2 = 10.$$

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

When a variable is given a simple value, substitute it before doing any algebra. A negative exponent just flips the number, so $\left(\frac12\right)^{-1}$ becomes the whole number $2$. Also remember: dividing by a number smaller than $1$ makes the answer bigger, not smaller.

Simplify first, then substitute. The top is $2a^{-1} + \frac12 a^{-1} = \frac52 a^{-1}$. Dividing by $a$ gives $\frac52 a^{-2}$. With $a = \frac12$, $a^{-2} = 2^2 = 4$, so the value is $\frac52 \cdot 4 = 10$.

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 $\boxed{\textbf{(D)}\ 10}$.