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```Date: 06/11/2003 at 04:17:28
From: Rachelle
Subject: Contest

There are 30 multiple-choice questions in a contest. Five marks are
awarded for each correct answer, and three marks are deducted for each

If a student scores 78 marks, what is the greatest possible number of
```

```
Date: 06/11/2003 at 12:57:55
From: Doctor Peterson
Subject: Re: Contest

Hi, Rachelle.

Here is one approach that is not too hard.

Let's say you get X right and Y wrong; then your score is 5X-3Y. To
get a score of 78, you must have

5X - 3Y = 78

If you know X, you can find Y by

5X = 3Y + 78

5X - 78 = 3Y

5/3 X - 26 = Y

Now you can make a table showing what Y is for different values of X,
starting with 30, the highest possible value:

X   Y = 5/3 X - 26
----  --------------
30   50-26 = 24
...

Clearly X can't be 30, because X+Y is greater than 30, the total
number of questions. You want to continue the table for smaller values
of X until X+Y is no more than 30.

You will quickly notice that you don't need to work out every line of
the table. What is the next value of X to try, that will give a whole
number for Y? How can you quickly fill in Y for the next row? Thinking
like this is great for discovering your own shortcuts.

You will find the answer pretty quickly this way; but even then you
may notice that you really didn't have to fill in the table at all.
There is another equation (or inequality) you could solve to find the
answer. But since it only took a few rows of the table to find the
answer, there's nothing wrong with not doing it the quickest way. You
may find a future problem where it does make a difference, though, so
it's a good idea to try solving the problem a second time using what
you learned, so that you will have a new technique for future use.
That's how mathematicians are made: we learn not only from our
mistakes, but from looking for alternative solutions.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra
Middle School Word Problems

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