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Where Do the Parentheses Go?

Date: 09/12/2003 at 00:48:21
From: Lynn
Subject: order of operations and writing equations

I'm trying to help my daughter with some problems she's been given. 
Each one has a string of numbers and operations on one side, and a
result on the other side.  What you're supposed to do is insert
parentheses around the numbers and operations in order to get the
result.  For example:

      2    2    2
 2 + 7  - 3  / 3  - 1 * 5 = 35

I know how PEMDAS works, but I don't see how to solve these problems
without just doing a lot of guessing.  Argh!


Date: 09/12/2003 at 08:53:18
From: Doctor Peterson
Subject: Re: order of operations and writing equations

Hi, Lynn.

This sort of problem is just a puzzle -- there is no standard, 
straighforward way to solve it, you just have to try things out and 
make intelligent guesses. It may help to read what we have to say 
about the order of operations

  http://mathforum.org/library/drmath/sets/select/dm_order_op.html 

but there is no specific technique we can give you. I'll just try to 
get you started, thinking through the problem until I see where to go.

Using our e-mail notation, your equation is

  2 + 7^2 - 3^2 / 3^2 - 1 * 5 = 35

Looking at this, I notice that 35 = 5*7, and that there is a 7 and a 
5 in the left side. So my first action is to add parentheses so that 
multiplication by 5 will be the last operation performed; if we can 
make the rest of the expression equal 7, we'll be in good shape. This 
is just a guess; it may turn out that we won't want to multiply by 5 
at all. But we can try it:

  (2 + 7^2 - 3^2 / 3^2 - 1) * 5 = 35

Now, it's not obvious that we can get 7 out of that, or that the 7 
that is there will in any way show through in the final form (since 
we can't undo the squaring, and we would have to divide by 7 to leave 
ourselves with just one 7). But I do see, at least, that 7^2 is much
too big, so we have to make it smaller, either by subtracting
something big or by dividing. 

As written, the division only affects 3^2; 3^2/3^2 is 1. So we'll want
to have parentheses around at least part of 

  2 + 7^2 - 3^2

and we can decide eventually how much of 

  3^2 - 1

to divide by. If we used all of both, we'd have

  (2 + 7^2 - 3^2) / (3^2 - 1) = 42/10

which is in the right ballpark but not a whole number, much less 
exactly 7. It doesn't help if we divide 42 by 3^2 or by 3 rather than 
3^2 - 1.

Now we can try either adding more parentheses inside (2 + 7^2 - 3^2) 
to change what we are dividing, or pull the 2 outside of the 
parentheses.

At this point I've finally looked ahead enough to see the solution. 
I'll leave you with just a hint: you'll want to take the first choice 
I mentioned in the last paragraph, and you'll have to change what you 
divide by as well. See what you can do.

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

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Puzzles
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