2. The Intermediate Maths Challenge has 25 questions with the following scoring rules:
5 marks are awarded for each correct answer to Questions 1-15;
6 marks are awarded for each correct answer to Questions 16-25;
Each incorrect answer to Questions 16-20 loses 1 mark;
Each incorrect answer to Questions 21-25 loses 2 marks.
Where no answer is given 0 marks are scored.
Fiona scored 80 marks in total. What possible answers are there to the number of questions Fiona answered correctly?
Solution
As each question is worth at most 6 marks, it is not possible to get more than $13 \times 6=78$ marks by correctly answering 13 (or fewer) questions.

If Fiona answered correctly 18 questions, the lowest number of marks she could get for her correct answers would be $15 \times 5+3 \times 6=93$ and the highest number of marks she could lose in the remaining seven questions would be $5 \times 2+2 \times 1=12$. Hence by answering 18 (or more) questions correctly, Fiona would always score at least $93-12=81$ marks.

Fiona could have answered correctly $14,15,16$ and 17 questions, for example:
- 14: correct answers to questions $12-25(4 \times 5+10 \times 6)$,
- 15: correct answers to questions $6-20(10 \times 5+5 \times 6)$,
- 16: correct answers to questions 1 - 16 and an incorrect answer to question 17,
- 17: correct answers to questions $1-17$ and incorrect answers to questions $20-23$.