Printable

2007 AMC 10B Problem 6

2007 AMC 10B Problem 6AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P1-10

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?

A)
$13$
B)
$14$
C)
$15$
D)
$16$
E)
$17$

Answer

Key insight

The three blanks bank 4.5 points, so she needs 6c >= 95.5, and the smallest whole number of correct answers is 16.

Solution

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

Why this works

Scoring rules are linear functions of the number correct, so the question is a single inequality. The only care needed is at the end: a fractional threshold in a "how many must" question is always rounded up.

The trap

Rounding 95.5/6 = 15.9 down to 15, which scores only 94.5 and falls short of 100.

Common mistakes

Rounding 95.5/6 = 15.9 down to 15, which scores only 94.5 and falls short of 100.
Forgetting the $4.5$ points from the blank problems and solving $6c \ge 100$ , which gives $17$ .

Techniques

Set up the equation/formula and compute; no special trick needed