How could I find all the possible scores for a test given that correct answers are 1 point, skipped answers are 0, & wrong answers are -0.25 points?

Question

I am creating scoreboards for a test that has 3 sections (40, 50 and 60 questions).

Correct answers count for 1 point, meaning the highest scores are 40, 50 and 60 respectively.

Skipped answers count for 0 points.

Wrong answers have a "penalty" of -0.25.

I would like to create a list of all the possible scores that someone could get on each section. For the 40 question section, 40/40 is possible, while 39.75/40 is not (since if you got 39 correct and 1 wrong, you'd actually be at 38.75).

I am not sure whether to be thinking of skipped questions as "plus zero" or as "minus one", and similarly I am not sure whether to think of wrong answers as "minus 0.25" or "minus 1.25". I believe both ideas work but am I confusing myself with which direction to think as I try to make a list of all possible scores.

I tried to make a matrix where x = the number of correct answers, y = the number of wrong answers, z = the number of skipped answers and S = your final score.

This is what I came up with:

x - 0.25y = S x + y + z = 40 (or 50, or 60, depending on the section) x, y and z are integers between 0 and 40 (or 50, or 60, depending on the section)

Solving this matrix yields

I don't know where to go from here. I tried to solve the list of all possible scores manually but am getting quickly confused and would like a method to how to be sure I'm getting all the possible scores and eliminated all the impossible ones.

Thanks a lot!

Answer 1

Let $C$ be the number of correct answers, $W$ the number of wrong answers, $Q$ the total number of questions in the section, and $S$ the final score multiplied by 4 (just to keep everything in integers, it doesn't really affect anything). Then:

1. $C, W \geq 0$
2. $C + W \leq Q$
3. $S = 4C - W$

Applying some fairly loose limits, we can quickly see that $-Q \leq S \leq 4Q$ is the only range where we might be able to achieve solutions.

Then we can note that with $C = 0$ you get $S = -W$ and $0 \leq W \leq Q$, and so you can achieve every possible score $S = -Q, \ldots, 0$. For positive scores, let $W = 4A + B$, where $0 \leq B \leq 3$. Then we get that $S = 4C - W = 4C - (4A + B) = 4(C - A) - B \leq 4C - B$. Notice that then if $A \neq 0$, we can replace $W$ with $B$ and $C$ with $C - A$ to achieve the same score, and so if we can't achieve a particular score by setting $A = 0$ then we can't achieve it at all. (There's a little bit of algebra hidden in all of that that I leave as an exercise.)

So then the valid positive scores are the ones that we can achieve when $W = 0, 1, 2, 3$, and $C = 1, 2, \ldots, Q - W$. Running through each of those gives:

$W$	$C$	$S$
$0$	$1, 2, \ldots, Q$	$4, 8, \ldots, 4Q$
$1$	$1, 2, \ldots, Q - 1$	$3, 7, \ldots, 4Q - 5$
$2$	$1, 2, \ldots, Q - 2$	$2, 6, \ldots, 4Q - 10$
$3$	$1, 2, \ldots, Q - 3$	$1, 5, \ldots, 4Q - 15$

In other words, in addition to every non-positive score from $-Q$ to $0$, we can achieve every positive score from $1$ to $4Q$ with 6 exceptions: $4Q - 11$, $4Q - 7$, $4Q - 6$, $4Q - 3$, $4Q - 2$, and $4Q - 1$.

Answer 2

Hint： $$40-((1-0)x+(1-(-0.25))y)$$ where $x, y$ are nonnegative integers and $$x+y \le 40$$