1986 AHSME Problems/Problem 12

Problem

John scores $93$ on this year's AHSME. Had the old scoring system still been in effect, he would score only $84$ for the same answers. How many questions does he leave unanswered? (In the new scoring system that year, one receives $5$ points for each correct answer, $0$ points for each wrong answer, and $2$ points for each problem left unanswered. In the previous scoring system, one started with $30$ points, received $4$ more for each correct answer, lost $1$ point for each wrong answer, and neither gained nor lost points for unanswered questions.)

$\textbf{(A)}\ 6\qquad \textbf{(B)}\ 9\qquad \textbf{(C)}\ 11\qquad \textbf{(D)}\ 14\qquad \textbf{(E)}\ \text{Not uniquely determined}$

Solution

Let $c$ , $w$ , and $u$ be the number of correct, wrong, and unanswered questions respectively. From the old scoring system, we have $30+4c-w=84$ , from the new scoring system we have $5c+2u=93$ , and since there are $30$ problems in the AHSME, $c+w+u=30$ . Solving the simultaneous equations yields $u=9$ , which is $\boxed{B}$ .

1986 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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1986 AHSME Problems/Problem 12

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