Where to Put the Parentheses?

Date: 01/03/2002 at 20:44:23
From: Hennaysha Candler
Subject: Order of operations

I am stuck on where to put parentheses in a math expression to make 
the expression true. I tried to use the guess-and-check strategy but 
it wouldn't work for me. Here is a problem I need to put parentheses 
into to make the equation true:

   9 - 6 + 4 * 6 / 3 = -2

Date: 01/03/2002 at 23:04:01
From: Doctor Peterson
Subject: Re: Order of operations

Hi, Hennaysha.

I can't think of any method you can use other than guess and check; 
you just have to come up with the right guess. That makes it hard for 
me to come up with a good hint other than giving you the answer. But 
I'll go through the problem to illustrate how you might think.

You have a negative answer, but there's only one negative sign in the 
whole expression. If you just subtract 6 from 9, you'll get 3, and 
there will be no way to make a negative answer, so you must be 
subtracting some expression greater than the 6 alone from the 9:

   9 - (6 + 4 * 6 / 3 = 3

But where can we put the right parenthesis? We can try all possible 
places:

   9 - (6 + 4) * 6 / 3 = 3  ==>  9 - 10 * 6 / 3
   9 - (6 + 4 * 6) / 3 = 3  ==>  9 - 34 / 3
   9 - (6 + 4 * 6 / 3) = 3  ==>  9 - 14

None of these gives -2; but looking back at them, I see that I can 
place another pair of parentheses in the first attempt and get the 
right answer:

   [9 - (6 + 4)] * 6 / 3 = 3  ==>  [9 - 10] * 6 / 3 = -1 * 6 / 3 = -2
    
The best "guess-and-check" works like this: you look for ways to 
restrict your guesses, make them, and then refine those based on the 
results. In this case, it worked out well to actually simplify each 
parenthesized version and then repeat the process by looking for 
places to add another pair; there probably won't always be such an 
obvious choice for the first pair, but this idea may help at least a 
little. Mostly it takes patience; if you haven't guessed, this kind of 
exercise is meant to give you LOTS of practice evaluating expressions 
with parentheses, so you can expect to have to do a lot of checking.

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