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.

6. Addition and subtraction.

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/