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### Order of Operations without Parentheses

```
Date: 8/3/96 at 10:15:1
From: Ken Dunham
Subject: Order of Operations for Unstructured Problem...

If given a problem with no parenthesis to indicate the order of
operations you're suppose to do * and / first, then + and -.  But how
do you solve a problem like:

21+82-59+29*86*66/50*7/87

The first part of the problem is 44 - rewritten as:

44+29*86*66/50*7/87

But how do you deal with all the multiplication and extra division
symbols?  My computer came up with the answer of 309 (rounded off).
Do you solve it like:

44+([29*86*66]/50)*(7/87)

or

44+(29*86)*[(66/50)*(7/87)]

or what?  How do you determine the order of operations for such a
crazy problem?
```

```
Date: 8/6/96 at 10:52:54
From: Doctor Jerry
Subject: Re: Order of Operations for Unstructured Problem...

You may have noticed that the last two expressions have the same
value, 7722/25, when they are evaluated by a calculator.  This agrees
with the rounded result you gave above.

You may find the following extract from the HP48G calculator User's
Guide useful:

(The order of operations, from highest to lowest, is given below.
Functions with the same precedence are performed from left to right.)

1. Expressions within parentheses.  Expressions within nested
parentheses are evaluated from inner to outer.

2. Prefix functions (such as sin, ln, ...).

3. Postfix functions (such as ! (factorial)).

4. Power (^) and square root.

5. Negation (-), multiplication, and division.

Applying these rules to 21+82-59+29*86*66/50*7/87 gives

((21+82)-59)+(((((29*86)*66)/50)*7)/87)

Not easy to read, but the parentheses force arithmetic on pairs of
numbers. So, first calculate 29*86 = 2494.  Then 2494*66, etc.

Hope this helps.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra

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