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 each unanswered question, then a student 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?
Step-by-step Solution:
Let \( c \), \( w \), and \( u \) represent the number of correct, wrong, and unanswered questions respectively. We know the total number of questions: \[ c + w + u = 160 \] Based on the first scoring condition: \[ c - 0.25w - 0.5u = 79 \] (Equation 1) Based on the second scoring condition: \[ c - 0.5w - 0.25u = 76 \] (Equation 2) Subtract Equation 2 from Equation 1: \[ 0.25w - 0.25u = 3 \implies w - u = 12 \implies w = u + 12 \] Substitute \( w \) in the total questions equation: \[ c + (u + 12) + u = 160 \implies c + 2u = 148 \implies u = \frac{148 - c}{2} \] Substitute \( w = u + 12 \) into Equation 1: \[ c - 0.25(u + 12) - 0.5u = 79 \] \[ c - 0.75u - 3 = 79 \implies c - 0.75u = 82 \] Substitute \( u = \frac{148 - c}{2} \): \[ c - 0.75\left(\frac{148 - c}{2}\right) = 82 \] \[ c - 0.375(148 - c) = 82 \] \[ c - 55.5 + 0.375c = 82 \] \[ 1.375c = 137.5 \implies c = 100 \]