Multiple choice
How many questions did Sarah answer correctly in this multiple choice exam?
Age
11 to 14
Challenge level
Problem
In a multiple choice exam, each candidate scores 4 points for every correct answer and loses 1 point for every incorrect answer.
Sarah answers all 100 questions and gets a score of 210 marks. How many questions did she answer correctly?
Student Solutions
Answer: 62
Using trial and improvement
50 correct: 200 points
50 incorrect: - 50 points
total: 150 points, too small
75 correct: 300 points
25 incorrect: - 25 points
total: 275 points, too big
60 correct: 240 points
40 incorrect: - 40 points
total: 200 points, too small
62 correct: 248 points
38 incorrect: - 38 points
total: 210 points, perfect
Starting from 100 correct
100 correct: 400 points
Each question incorrect instead of correct: Lose 4 points for not correct
And 1 more for incorrect
Total lose 5
400 - 210 = 190
190$\div$5 = 38
38 questions were incorrect and not correct
The other 62 were correct
Using coding
If 1 is added to the score for each question, then, for candidates who answer all of the questions, then the scoring system becomes 5 points for each correct answer and none for each incorrect answer.
Adding 1 to the score for each question is the same as adding 100 to the total score, so under the scoring system of 5 points for each correct answer and none for each incorrect answer, Sarah would have got 210 + 100 = 310 points.
310$\div$5 = 62, so Sarah answered 62 questions correctly.
Using simultaneous equations
Suppose Sarah answered $s$ questions correctly and $t$ questions incorrectly. Then $s+t=100$, because there are $100$ questions altogether.
Sarah would score $4s$ points for the $s$ correct answers, and lost $t$ points for the $t$ incorrect answers, so she would score $4s-t$ points. So $4s-t=210$.
Adding these two equations gives $s+t+4s-t=100+210 \Rightarrow 5s=310 \Rightarrow s=62$.