Questions Answered Correctly

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 
incorrect answer. No marks are awarded for questions left unanswered. 

If a student scores 78 marks, what is the greatest possible number of 
questions answered correctly?

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/