Algebra help.

Hi, I'm new here just signed up. I got a problem for homework that's been troubling me so I did a quick google for math help forums and came up with this place. Any help you give me is MUCH appreciated, I also hope I posted this in the right area.

Anyway here is the question:

Peter entered a competition in which he got 5 points for every correct answer, but lost 3 points for every wrong answer. At the end of the competition he had no points at all and had answered 16 questions.

That was basically the first statement. Then we had to work out this problem:

Work out how many questions Peter got right from the 16 questions he answered.

I did this correctly, I think.

X questions wrong @ -3 points ea.
16 - X questions right @ 5 points ea.

-3x + 5 (16 - x) = 0

-3x + 80 - 5x = 0

-8x = -80

x = 10.

I'm fairly sure that I got that question right, but further down the track came this question;

What numbers of questions could you answer to end up with no points?

I thought that I could use the same formula in a way as the first question, so I used this:

Let there be N questions where 0 > n > 20
Let the number of correct answers = X @ 5 points ea.
Therefore, let the number wrong = n - x. @ -3 points ea.

5x - 3 (n - x) = 0

Now I'm a bit confused. If I keep working it out I'll get something like this:

5x - 3n - 3x = 0

2x - 3n = 0

Now I guess I should take one of these to the other side of the equation.

2x = 3n

Now I'm totally stumped. The teacher said you'd have to use some guesswork to get the answer, but I have no idea how you could do that either.

Aha, I think I've got it. I think I just have to make n and x numbers and find out how many different combinations make 0.

Is this right?

But how do I find N? By replacing X with numbers or something?

:?

Damnit, I feel like such an idiot.

I worked the whole thing out. Basically you just replace N with a number between 1 and 20 and see if it's divisable in the end. For example 16 questions works because you can get 10 right and 6 wrong. So 30 - 30 = 0.

5x - 3 (16-x) = 0
5x - 48 + 3x = 0
8x - 48 = 0
8x = 48
x = 6

It didn't turn out as a fraction so therefore with sixteen questions you can get a score of zero.

I'm so happy I worked it out, I think I was misreading the question.

Bea, the relationship is 8x = 3n

So (as I said earlier), x MUST be a multiple of 3;
so simply substitute for x (much easier):
x=3: 3n = 8(3) ; n = 8
x=6: 3n = 8(6) ; n = 16
...and so on: solutions are infinite

But my solution is correct as well isn't it? As long as X equals a whole number at the end you can end up with a score of zero.

You said n=1 to 20 so I assume (though you didn't say so) that it is a 20 question test. So
r=right, w=wrong, s=skipped.
r+w+s=20 =
5r+5w+5s=100 and
5r-3w+0s=0
Subtracting
8w+5s=100 or
s=20-8w/5 so w is a multiple of 5.
The w possibilities are 0,5 & 10 to avoid skipping a negative number. That's all!