5.
AHSME
The Mathematical Association of America sponsors a high school
mathematics contest every year called the American High School
Math Exam. The contest consists of multiple-choice questions that
include five possibilities for each question. For many years, the scoring
system for the 30-question test was as follows: Each right answer was
worth 4 points. Each wrong answer was worth \(-1\) point. Each
unanswered question was worth zero points. Each contest participant
started with a score of 30 points. Thus, for a person who got 6 right,
3 wrong, and left 21 unanswered, the score would be $30 + (6 \times 4)$
$\+ (3 \times -1) + (21 \times 0) = 51$ points. To qualify for the second round of
competition, a person had to score 95 points.
Around 1988 the scoring system changed. The new scoring system
is as follows: 5 points for a right answer, zero points for a wrong
answer, 2 points for no answer, and each participant starts with zero
points. Thus, the example of 6 right, 3 wrong, and 21 unanswered
would result in $(6 \times 5) + (3 \times 0) + (21 \times 2) = 72$ points. To qualify
for the second round of competition, a person has to score 100 points.
Analyze these two different scoring systems and discuss which
system you think gives a person a better chance of qualifying for
the next round. For each system, decide on the best strategy of
reaching the second round, given that you are sure of the answers to
10 questions and have narrowed down 10 other questions to two
choices.
