Reba's Physics Problems

Date: 12/09/2001 at 18:52:15
From: Leslie Matthews
Subject: Algebra equation

To motivate Reba to work her physics problems correctly, her dad said 
he'd pay her a quarter for each correct answer and fine her a dime 
for each incorrect answer. If she received $3.80 after doing 25 
problems, how many problems did Reba answer correctly?

Date: 12/10/2001 at 10:46:25
From: Doctor Ian
Subject: Re: Algebra equation

Hi Leslie,

Well, did she get all 25 correct?  We can check that.  

    25 correct * 25 cents - 0 incorrect * 10 cents

  = 25^2 - 0

  = 625 cents

So she must have missed at least some.  

Now that we know how to _check_ an answer, we have some options. One 
option is guessing. Did she get 10 questions right? Let's check:

    10 correct * 25 cents - 15 incorrect * 10 cents

  = 250 - 150

  = 100 cents


So that's too low. In fact, we can try to be smart about guessing by 
thinking ahead of time about what guesses can't possibly be right. In 
this case, we know that she has to get at least $3.50, which is 14 
quarters, so there is no point in guessing that she got fewer than 14 
answers correct. So now we know the answer must be between 15 and 25 - 
we've eliminated half the possibilities right there.

We can also be clever by realizing that, if we start from the best 
case of getting 25 correct, each time she blows a question it costs 
her 35 cents: the 25 cents she would have got, plus the 10-cent 
penalty. So missing 1 question would give her

  625 - 1*35 = 590 cents

Missing two questions would give her

  625 - 2*35 = 555 cents

and so on. You can keep trying values for the number of questions 
missed, and eventually you should arrive at one that gives you a total 
of 380 cents.  

Of course, you can be even more clever by realizing that this is 
really an equation with a variable:

  625 - n*35 = 380

If you can solve this for n, you can find out immediately how many 
questions she must have missed.  

Finally, we can try to solve this directly 'by algebra'.  We have two 
pieces of information:

  1) The total number of questions (right and wrong) is 25:

         r + w = 25

  2) The total amount of money she ends up with is 380 cents:

       25r - 10w = 380

Once you have those equations, there is a variety of ways to go about 
proceeding directly to the answer. But of course, until you know how 
to tell whether whatever answer you come up with is correct or not, 
there's really not much point in cranking one out. 

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/