2007 AMC 10B Problem 6

P1-10~2 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

The $2007 \text{ AMC }10$ will be scored by awarding $6$ points for each correct response, $0$ points for each incorrect response, and $1.5$ points for each problem left unanswered. After looking over the $25$ problems, Sarah has decided to attempt the first $22$ and leave only the last $3$ unanswered. How many of the first $22$ problems must she solve correctly in order to score at least $100$ points?

Show solution without answering

Key insight

The three blank problems give 4.5 points for free, so the right answers must bring at least 95.5 more. Fifteen right is 90, too few; sixteen right is 96, enough.

Solution

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 $\boxed{\textbf{(D)}\ 16}$ .

Why this works

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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(D)}\ 16}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)