2007 AMC 10B Problem 6

1. The last $3$ problems are left blank. Each is worth $1.5$ points, so they give
$$3 \times 1.5 = 4.5 \text{ points}.$$
2. Wrong answers score $0$, so the rest of her score comes only from right answers, $6$ points each.
3. She needs at least $100$ points in all, so the right answers must give at least
$$100 - 4.5 = 95.5 \text{ points}.$$
4. Try numbers of right answers near $95.5 \div 6$:
- $15$ right: $15 \times 6 = 90$. Total $90 + 4.5 = 94.5$. Not enough.
- $16$ right: $16 \times 6 = 96$. Total $96 + 4.5 = 100.5$. Enough.
5. So the smallest number that works is $16$.

The answer is $\boxed{\textbf{(D)}\ 16}$.

The score grows by the same $6$ points for each extra right answer, so you only need one inequality. The care comes at the end: you cannot solve part of a problem, so when the division gives $15.9\ldots$, the number of problems must be rounded up. Checking the two neighbors, $15$ and $16$, confirms it.

The three unanswered problems are worth $3 \cdot 1.5 = 4.5$ points no matter what. Wrong answers score nothing, so if Sarah gets $c$ of the attempted problems right her score is $6c + 4.5$.

She needs
$$6c + 4.5 \ge 100 \quad\Longrightarrow\quad 6c \ge 95.5 \quad\Longrightarrow\quad c \ge 15.9\ldots$$
Since $c$ is a whole number, the smallest value that works is $c = 16$. Check: $16 \cdot 6 + 4.5 = 100.5 \ge 100$, while $15$ correct gives only $94.5$.

The answer is $\boxed{\textbf{(D)}\ 16}$.