Question

An examination consists of 160 questions. One mark is given for every correct option. If a 1/4 mark is deducted for each wrong option and a half mark is deducted for every unanswered question, the person scores 79. If half a mark is deducted for every wrong option and 1/4 mark is deducted for every unanswered question, the person scores 76. Find the number of correct answers he wrote?

• A. 80
• B. 100
• C. 120
• D. 140

Hint:

Using substitution method for three variables often results in simpler arithmetic than elimination.

Answer 1

The Correct Option is B

Step 1: Define the three counts.
Let $C$, $W$, $U$ be the numbers of correct, wrong and unanswered questions, with $C + W + U = 160$.
Step 2: Write the first scoring scheme.
Plus 1 for correct, minus $\tfrac14$ for wrong, minus $\tfrac12$ for blank gives $C - 0.25W - 0.5U = 79$.
Step 3: Write the second scoring scheme.
Now minus $\tfrac12$ for wrong and minus $\tfrac14$ for blank gives $C - 0.5W - 0.25U = 76$.
Step 4: Subtract to relate W and U.
Subtracting the second from the first: $0.25W - 0.25U = 3$, so $W - U = 12$.
Step 5: Reduce to one unknown.
Add the two scoring equations: $2C - 0.75W - 0.75U = 155$, i.e. $2C - 0.75(W+U) = 155$. Since $W + U = 160 - C$, this becomes $2C - 0.75(160 - C) = 155$.
Step 6: Solve for C.
Expand: $2C - 120 + 0.75C = 155$, so $2.75C = 275$ and $C = 100$.
\[ \boxed{100} \]