I have the following problem solving question:
In a multiple choice test of 25 questions, four marks are given for each correct answer and two marks are deducted for each wrong answer. One mark is deducted for any question which is not attempted. James scores 55 marks and wants to know how many questions he got right. He can't remember how many questions he did not attempt, but he doesn't think it was very many. How many questions did James get right?
I've worked out that he gained $2.2$ marks per question on average and lost $1.8$ marks per question on average (because $55/25$ is $2.2$ and $4-2.2$ is $1.8$).
I've also stated that $2x+y=45$, where $x$ is the number of questions he got wrong and $y$ is the number of questions he didn't answer. Consequently, $100-(2x+y)=55$.
Additionally, I know that getting every question wrong would result in $-50$ total marks, not answering any question would result in $-25$, and answering all correctly would result in $100$.
However, I don't know what to do with any of this information.
From the information about the test and knowledge that he earned 55 points, we have $$ 55 = 4(25-y-x)-2x-y=100-4y-4x-2x-y=100-6x-5y $$ or equivalently, $$ 6x+5y=45. $$ We know that $y$ and $x$ must each be nonnegative integers, and that $y$ must be "small". Setting $y=0$, $y=1$, and $y=2$ gives $x=45/6$, $x=40/6$, and $x=35/6$ respectively, which aren't valid because these are integer values. However setting $y=3$ yields $x=30/6=5$.
Thus James answered 22 questions, got 5 wrong, and 17 correct. To check our answer, observe that $$4(17)-3-2(5)=68-3-10=55.$$ The only other possible answer is to have $y=9$ and $x=0$, which says that James answered 16 questions, got 0 wrong, and 16 correct.