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### Where to Put the Parentheses?

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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.

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

[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/
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